# Horizon thermodynamics in fourth-order gravity

## TL;DR

This paper demonstrates that horizon thermodynamics can be extended to fourth-order gravity by transforming it into a second-order form, allowing the field equations to be expressed as thermodynamic identities and deriving consistent black hole masses.

## Contribution

The paper introduces a Legendre transformation to convert fourth-order gravity into a second-order form, enabling the construction of horizon thermodynamics in higher-order gravity theories.

## Key findings

- Field equations in fourth-order gravity can be written as thermodynamic identities.
- The approach yields the same black hole mass as other methods.
- Horizon thermodynamics extends to higher-order gravity theories.

## Abstract

In the framework of horizon thermodynamics, the field equations of Einstein gravity and some other second-order gravities can be rewritten as the thermodynamic identity: $dE=TdS-PdV$. However, in order to construct the horizon thermodynamics in higher-order gravity, we have to simplify the field equations firstly. In this paper, we study the fourth-order gravity and convert it to second-order gravity via a so-called " Legendre transformation " at the cost of introducing two other fields besides the metric field. With this simplified theory, we implement the conventional procedure in the construction of the horizon thermodynamics in 3 and 4 dimensional spacetime. We find that the field equations in the fourth-order gravity can also be written as the thermodynamic identity. Moreover, we can use this approach to derive the same black hole mass as that by other methods.