May 28, 2026
The deep connection between gravity and thermodynamics has been a subject of intense investigation ever since the pioneering works of Bekenstein and Hawking, which revealed that black holes possess temperature and entropy proportional to their horizon area. This insight was later extended to cosmological settings, where it was shown that the Friedmann equations governing the expansion of the universe can be derived from the first law of thermodynamics applied to the apparent horizon [1]–[4]. Such a gravity-thermodynamics correspondence suggests that gravitational field equations may be interpreted as a thermodynamic equation of state, opening a new window to understand the fundamental nature of gravity [5]–[9].
In parallel, modified gravity theories have gained significant attention as they offer possible explanations for the late-time accelerated expansion of the universe without invoking dark energy, as well as for early-universe inflation. Among these, curvature based modifications such as \(f(R)\) gravity have been extensively studied [10]–[13]. However, less explored are modifications that involve higher-order curvature invariants like the Kretschmann scalar \(R_{\rho \sigma \mu \nu} R^{\rho \sigma \mu \nu}\) which naturally appear in quantum gravity corrections and string theory effective actions.
In this work, we propose a novel modified gravity action for a spatially flat Friedmann-Robertson-Walker (FRW) universe, where the Einstein-Hilbert term is supplemented by an arbitrary function \(f\) of a particular combination of the Ricci scalar and the Kretschmann invariant, namely, \(f=f(\mathcal{R}+\sqrt{6\mathcal{R}_{\rho \sigma \mu \nu }\mathcal{R} ^{\rho \sigma \mu \nu }-\mathcal{R}^{2}})\). This combination is chosen for its geometric simplicity and to avoid introducing higher-order derivatives in the resulting field equations. The action contains an extension parameter \(\lambda\), and when \(\lambda=0\) standard general relativity is recovered.
We derive the modified Friedmann equations and then, following the gravity- thermodynamics conjecture, study the thermodynamic behavior of the apparent horizon. By employing the first law of thermodynamics along with the continuity equation and the modified Friedmann equations, we obtain a general expression for the entropy associated with the apparent horizon. This entropy reduces to the standard area-law entropy in the limit \(\lambda\rightarrow 0\) and can be explicitly computed for various well-known entropy models by specifying the form of \(f\). Recently, deviations from the standard area-law entropy were obtained, using a different approach based on stochastic fluctuations of the spacetime metric [14]. Our results provide a consistent thermodynamic interpretation of the proposed modified gravity theory and may have implications for understanding the microscopic origin of cosmological entropy.
The paper is organized as follows. In Sec. 2, we present the modified gravity action and derive the corresponding modified Friedmann equations for a flat FRW universe. In Sec. 3, we investigate the thermodynamic properties of the apparent horizon, derive the modified entropy expression, and summarize results for several entropy models in a table. Finally, Sec. 4 provides our conclusions and outlook.
In this section, we propose a modified gravity action for the FRW universe and derive the corresponding modified Friedmann equations. We begin by considering the spatially flat line element of a homogeneous and isotropic metric in \((3+1)\)-dimensional spacetime, which is expressed as \[\label{Eq:FRW} ds^{2}=h_{ab}dx^{a}dx^{b}+R^{2}d\Omega ^{2},\tag{1}\] where \(x^{0}=t\), \(x^{1}=r\), \(R=a(t)r\), and \(a(t)\) is the scale factor. Here, \(h_{ab} = \text{diag}(-1, a^2(t))\) represents the metric of the two-dimensional \((t, r)\) subspace, and \(d\Omega ^{2}=d\theta ^{2}+\sin^2\theta d\phi ^{2}\) is the metric of a unit \(2\)-sphere.
Now, we generalize the standard Einstein-Hilbert action in \((3+1)\)-dimensional spacetime with an arbitrary function \(f\) as follows, \[\mathcal{S}=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left[ \mathcal{R}+\lambda f\!\left( \mathcal{R}+\sqrt{6\mathcal{R}_{\alpha \beta \mu \nu }\mathcal{R}^{\alpha \beta \mu \nu }-\mathcal{R}^{2}}\right) \right] +\mathcal{S}_{m},\] where \(\mathcal{R}\) and \(\mathcal{R}_{\alpha\beta\mu\nu}\) are the Ricci scalar and Riemann curvature tensor, respectively. Additionally, \(\lambda\) is an extension parameter and \(\mathcal{S}_m\) stands for the matter action. It is worth noting that the argument of the function \(f\) is constructed from a combination of the Ricci scalar and the Kretschmann scalar curvature invariants. By varying the above action with respect to the metric tensor \(g^{\mu\nu}\), we obtain the field equations in the following form: \[\label{Eq:EoM} G_{\mu \nu }+\lambda \,\mathcal{A}_{\mu \nu }=8\pi G\mathcal{T}_{\mu \nu },\tag{2}\] where \(G_{\mu\nu}\) and \(\mathcal{T}_{\mu\nu}\) are the Einstein and stress-energy tensors, respectively. We do not show the expression of \(\mathcal{A}_{\mu\nu}\) here, since it is rather lengthy and complicated. We specify the matter content of the universe as a perfect fluid, whose energy-momentum tensor is expressed as \[\mathcal{T}_{\mu \nu }=\left( \rho +p\right) u_{\mu }u_{\nu }+pg_{\mu \nu },\] where \(\rho\) and \(p\) are, respectively, the energy density and pressure, and \(u_{\mu}=(1,0,0,0)\) is the four-velocity of the fluid. The matter energy-momentum tensor is governed by the conservation law \(\nabla _{\mu }\mathcal{T}^{\mu \nu }=0\), which leads directly to the continuity equation \[\label{Eq:ConE} \dot{\rho}+3H\left( \rho +p\right) =0,\tag{3}\] where the overdot symbol means the derivative with respect to the cosmic time and \(H = \dot{a}/a\) represents the Hubble parameter. Hence, by substituting the flat FRW line element into the field equations 2 , the corresponding modified Friedmann equations take the following form \[\begin{align} \tag{4} &&H^{2}\left( 1-\frac{\lambda }{3}f^{\prime }\!\left( H^{2}\right) \right) +\frac{\lambda }{6}f\!\left( H^{2}\right) =\frac{8\pi G}{3}\rho, \\ &&\dot{H}\left( 1-\frac{\lambda }{6}f^{\prime }\!\left( H^{2}\right) -\frac{\lambda }{3}H^{2}f^{\prime \prime }\!\left( H^{2}\right) \right) =-4\pi G\left( \rho +p\right),\tag{5} \end{align}\] where the prime is the derivative with respect to \(H^{2}\). It should be noted that the resulting modified equations remain free of higher-order derivatives. Obviously, when \(\lambda = 0\), the standard Friedmann equations are recovered. In the next section, based on the gravity-thermodynamics conjecture, we shall establish the thermodynamic laws at the apparent horizon. By utilizing the conservation law along with the first modified Friedmann equation, we will then derive the entropy associated with this horizon.
In this section, based on the deep connection between gravity and thermodynamics, we intend to examine the thermodynamic behavior of the modified Friedmann equation and calculate the corresponding thermodynamic quantities at the apparent horizon boundary. In particular, we obtain the entropy expression associated with the apparent horizon of the FRW universe.
In the cosmological context, the radius of the dynamical apparent horizon is identified as the location where \(h^{ab}\partial _{a}R_{h}\partial _{b}R_{h}=0\). Consequently, the apparent horizon radius \(R_{h}\) for a spatially flat FRW universe is explicitly given by [15]–[18] \[\label{Eq:AH} R_{h}=1/H.\tag{6}\] The associated temperature with the apparent horizon is determined by \(T=\kappa /2\pi\), where \(\kappa\) is the surface gravity on the apparent horizon defined via [18] \[\kappa =\frac{1}{2\sqrt{-h}}\partial _{a}\left( \sqrt{-h}h^{ab}\partial _{b}R_{h}\right) .\]By using the FRW metric 1 , one obtains \[T=-\frac{1}{2\pi R_{h}}\left( 1-\frac{\dot{R}_{h}}{2HR_{h}}\right) . \label{Eq:Tem}\tag{7}\] Following the gravity-thermodynamics correspondence, we write down the first law of thermodynamics on the apparent horizon as \[dE=TdS_{h}+WdV_{h}, \label{Eq:FLT}\tag{8}\] where \(E=\rho V_{h}\) is the total energy in volume \(V_{h}=4\pi R_{h}^{3}/3\) enclosed by the apparent horizon, while \(T\) and \(S_{h}\) denote the temperature and entropy of the apparent horizon, respectively. Here, the quantity \(W\) is the work density associated with the volume change of the expanding universe given by [18] \[W=-\frac{1}{2}\mathcal{T}^{ab}h_{ab},\]which in terms of the energy density and pressure reads as \[W=\frac{1}{2}\left( \rho -p\right) . \label{Eq:Work}\tag{9}\]
| Entropy model | Modified Lagrangian | \(S_{h}\) |
| Kaniadakis [19]–[21] | \(\mathcal{R}-\lambda X^{-1}\) | \(S+\frac{\lambda G^2}{72\pi^2} S^3\) |
| Rényi [22], [23] | \(\mathcal{R}-\lambda \ln(X)\) | \(S-\frac{\lambda G }{12\pi}S^2\) |
| Logarithmic [24], [25] | \(\mathcal{R}-\lambda X^{2}\) | \(S+\frac{ 144\pi \lambda}{G} \ln(S)\) |
| Inverse-area [26] | \(\mathcal{R}+\lambda X^{3}\) | \(S+\frac{4320\pi^2 \lambda}{G^2}S^{-1}\) |
| Barrow [27], [28] | \(\mathcal{R}-\lambda X(4-\ln(\frac{XG}{12\pi}))\) | \(S+2\lambda S\ln(S)\) |
| MOND [29] | \(\mathcal{R}-\lambda X^{1-n}\) | \(S+\frac{2^{1-2n}(1-n)(1-2n)\lambda }{3^n(1+n)\pi^{n}G^{-n}} S^{1+n}\) |
We now proceed to derive the entropy of the apparent horizon based on the modified Friedmann equations 4 . Taking the differential of the total energy inside the apparent horizon, one obtains \[\begin{align} dE& =\dot{\rho}V_{h}dt+\rho dV_{h}, \notag \\ & =-4\pi R_{h}^{2}\left[ p+\rho (1-\dot{R}_{h})\right] dt, \end{align}\]where the continuity equation 3 has been used in the second equality. On the other hand, the first law of thermodynamics 8 together with the expressions for the temperature 7 and the work density 9 yields \[dE=-\frac{1}{2\pi R_{h}}\left( 1-\frac{\dot{R}_{h}}{2}\right) dS_{h}+2\pi R_{h}^{2}\dot{R}_{h}\left( \rho -p\right) dt.\]Combining the above expressions for the energy differential leads to \[dS_{h}=8\pi ^{2}R_{h}^{3}\left( \rho +p\right) dt,\]which, upon using the continuity equation 3 , reduces to \[dS_{h}=-\frac{8}{3}\pi ^{2}R_{h}^{4}d\rho . \label{Eq:Sh}\tag{10}\] Next, rewriting the first Friedmann equation 4 in terms of the apparent horizon radius \(R_{h}\) using 6 and taking its differential yields \[-2R_{h}^{-3}\left( 1-\frac{\lambda }{6}f^{\prime }\!\left( R_{h}^{-2}\right) -\frac{\lambda }{3}R_{h}^{-2}f^{\prime \prime }\left( R_{h}^{-2}\right) \right) dR_{h}=\frac{8\pi G}{3}d\rho ,\]where the prime is the derivative with respect to \(R_{h}^{-2}\). Finally, substituting the above result into 10 and integrating, we find the modified entropy of the apparent horizon as \[S_{h}=S-\frac{\lambda G}{6\pi }\int S^{2}\left[ 3f^{\prime }(S)+2Sf^{\prime \prime }(S)\right] dS,\]where \(S=\pi R_{h}^{2}/G\) is the standard entropy proportional to the horizon area and the prime denotes differentiation with respect to \(S\). Note that the integration constant is set to zero to recover the standard area-law entropy when \(\lambda =0\). Thus, for a given functional form of \(f\), the corresponding modified entropy of the apparent horizon can be explicitly determined. The functional forms of \(f\) corresponding to several well-known entropy models are summarized in Table 1.
In this work, we have proposed a modified gravity theory in a spatially flat FRW universe by extending the standard Einstein-Hilbert action with an arbitrary function of the Ricci scalar and the Kretschmann scalar invariant, specifically \(f=f(\mathcal{R}+\sqrt{6\mathcal{R}_{\rho \sigma \mu \nu }\mathcal{R}^{\rho \sigma \mu \nu }-\mathcal{R}^{2}})\). The inclusion of this particular combination ensures that the resulting modified Friedmann equations remain free of higher-order derivatives, a desirable feature for maintaining a well-defined initial value problem. The modified field equations contain an extension parameter \(\lambda\), and in the limit \(\lambda\rightarrow0\), the standard Friedmann equations of general relativity are recovered.
Using the gravity-thermodynamics conjecture, we examined the thermodynamic behavior of the apparent horizon of the FRW universe. By applying the first law of thermodynamics \(dE=TdS+WdV\) together with the continuity equation and the modified Friedmann equations, we derived a general expression for the entropy associated with the apparent horizon \[S_{h}=S-\frac{\lambda G}{6\pi}\int S^{2}\left[ 3f^{\prime }(S)+2Sf^{\prime \prime }(S)\right] dS,\] where \(S=\pi R_{h}^2/G\) is the standard Bekenstein-Hawking entropy associated with the apparent horizon. This expression explicitly shows how the modified gravity action leads to corrections to the area-law entropy. For a given functional form of \(f\), the corresponding modified entropy can be computed. The results for several well-known entropy models (e.g.,Rényi, Barrow, Kaniadakis, etc.) are summarized in Table 1, demonstrating the versatility of our approach.
Several interesting directions remain for future investigation. First, it would be valuable to explore the cosmological implications of the modified Friedmann equations derived here, such as the possibility of inflation or late-time acceleration without exotic matter components. Second, the explicit form of the tensor \(\mathcal{A}_{\mu \nu}\) in the field equations, though lengthy, could be analyzed for specific symmetric spacetimes to reveal additional physical effects. Third, extending this framework to include spatial curvature or to non-flat FRW universes would provide a more complete thermodynamic picture. Fourth, the connection between the modified entropy expression and quantum gravity corrections-such as those arising from loop quantum gravity or string theory-deserves further study. Finally, observational constraints on the parameter \(\lambda\) and the functional form of \(f\) could be derived from cosmological data, offering potential tests of the model. Our work thus provides a foundation for a deeper understanding of the interplay between modified gravity and the thermodynamics of spacetime.