Horizon thermodynamics in $f(R)$ theory

TL;DR

This paper examines the validity of the horizon first law in $f(R)$ gravity, deriving black hole entropy and energy formulas consistent with previous literature, and applies these to specific black hole solutions.

Contribution

It demonstrates that the horizon first law holds in $f(R)$ theories and derives new formulas for black hole entropy and energy within this framework.

Findings

Horizon first law is valid in $f(R)$ gravity.

Derived formulas for black hole entropy and energy.

Computed entropy and energy for specific $f(R)$ black hole solutions.

Abstract

We investigate whether the new horizon first law proposed recently still work in $f(R)$ theory. We identify the entropy and the energy of black hole as quantities proportional to the corresponding value of integration, supported by the fact that the new horizon first law holds true as a consequence of equations of motion in $f(R)$ theories. The formulas for the entropy and energy of black hole found here are in agreement with the results obtained in literatures. For applications, some nontrivial black hole solutions in $f(R)$ theories have been considered, the entropies and the energies of black holes in these models are firstly computed, which may be useful for future researches.