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

TL;DR

This paper investigates the laws of black hole thermodynamics within a modified gravity theory involving Ricci scalar, Ricci invariant, and scalar fields, deriving generalized thermodynamic laws and testing their validity with specific models.

Contribution

It formulates Friedmann equations in $f(R,R_{etaeta} R^{etaeta}, \, \phi)$ gravity as thermodynamic laws, extending previous work to more general theories.

Findings

Derived thermodynamic laws for apparent horizons in the modified gravity.

Expressed the generalized second law of thermodynamics in this context.

Produced parameter constraints for specific models to validate GSLT.

Abstract

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. Finally to check the validity of GSLT, we take some particular models and produce constraints of the parameters.