Thermodynamics in $f(R,R_{\alpha\beta}R^{\alpha\beta},\phi)$ theory of gravity

Understand

First and second laws of black hole thermodynamics are examined at the apparent horizon of FRW spacetime in $f(R,R_{\alpha\beta}R^{\alpha\beta} ,\phi)$ gravity, where $R$, $R_{\alpha\beta}R^{\alpha\beta}$ and $\phi$ are the Ricci scalar, Ricci invariant and the scalar field respectively.

• In this modified theory, Friedmann equations are formulated for any spatial curvature.
• These equations can be presented into the form of first law of thermodynamics for $T_{h}d\hat{S}_{h}+ T_{h}d_{i}\hat{S}_{h}+W dV=dE$, where $d_{i}\hat{S}_{h}$ is an extra entropy term because of the non-equilibrium presentation of the equations and $T_{h}d\hat{S}_{h}+W dV=dE$ for the equilibrium presentation.
• The generalized second law of thermodynamics (GSLT) is expressed in an inclusive form where these results can be represented in GR $f(R)$ and $f(R, \phi)$ gravities.