# Generalized Misner-Sharp energy in the generalized Rastall theory

## TL;DR

This paper derives a generalized Misner-Sharp mass and horizon entropy within the generalized Rastall theory, showing differences from Einstein and Rastall theories, and confirms the second law of thermodynamics in a flat FRW universe.

## Contribution

It introduces a new formulation of the Misner-Sharp mass and horizon entropy in the generalized Rastall theory, extending thermodynamic relations beyond Einstein gravity.

## Key findings

- The generalized Misner-Sharp mass differs from Einstein and Rastall theories.
- The horizon entropy matches the Rastall theory and reduces to Bekenstein entropy in Einstein limit.
- The second law of thermodynamics holds in the flat FRW universe under this framework.

## Abstract

Employing the unified first law of thermodynamics and the field equations of the generalized Rastall theory, we get the generalized Misner-Sharp mass of spacetimes for which $g_{tt}=-g^{rr}=-f(r)$. The obtained result differs from those of the Einstein and Rastall theories. Moreover, using the first law of thermodynamics, the obtained generalized Misner-Sharp mass and the field equations, the entropy of the static spherically symmetric horizons is also addressed in the framework of the generalized Rastall theory. In addition, by generalizing the study to the flat FRW universe, the apparent horizon entropy is also calculated. Considering the effects of applying the Newtonian limit to the field equations on the coupling coefficients of the generalized Rastall theory, our study indicates $i$) the obtained entropy-area relation is the same as that of the Rastall theory, and $ii$) the Bekenstein entropy is recovered when the generalized Rastall theory reduces to the Einstein theory. The validity of the second law of thermodynamics is also investigated in the flat FRW universe.