# Thermodynamics of $f(R)$ Gravity with Disformal Transformation

## TL;DR

This paper investigates the thermodynamic properties of $f(R)$ gravity under disformal transformations, analyzing the first and second laws in FLRW universe and revealing how disformal effects influence cosmic evolution.

## Contribution

It introduces a novel analysis of thermodynamics in $f(R)$ gravity with disformal transformations, including effects on the Hubble parameter and matter equations of state.

## Key findings

- Verification of generalized thermodynamic laws in disformal $f(R)$ gravity.
- Disformal terms induce effective variations in matter equations of state.
- Disformal effects modify the Hubble parameter in FLRW cosmology.

## Abstract

We study thermodynamics in $f(R)$ gravity with the disformal transformation. The transformation applied to the matter Lagrangian has the form of $\g_{\m\n} = A(\phi,X)g_{\m\n} + B(\phi,X)\pa_\m\f\pa_\n\f$ with the assumption of the Minkowski matter metric $\g_{\m\n} = \e_{\m\n}$, where $\phi$ is the disformal scalar and $X$ is the corresponding kinetic term of $\phi$. We verify the generalized first and second laws of thermodynamics in this disformal type of $f(R)$ gravity in the Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) universe. In addition, we show that the Hubble parameter contains the disformally induced terms, which define the effectively varying equations of state for matter.