# The zeroth law in quasi-homogeneous thermodynamics and black holes

## TL;DR

This paper explores the zeroth law of thermodynamics in systems with quasi-homogeneous entropy, especially black holes, proposing a revision to the law to reconcile phase coexistence and thermodynamic identities.

## Contribution

It introduces a revised zeroth law compatible with quasi-homogeneous entropy and applies it to black hole thermodynamics, challenging traditional thermodynamic assumptions.

## Key findings

- Generalized Gibbs-Duhem identity conflicts with standard zeroth law
- Proposed revision allows Maxwell's construction for black holes
- Application to black holes supports the revised thermodynamic framework

## Abstract

Motivated by black holes thermodynamics, we consider the zeroth law of thermodynamics for systems whose entropy is a quasi-homogeneous function of the extensive variables. We show that the generalized Gibbs-Duhem identity and the Maxwell construction for phase coexistence based on the standard zeroth law are incompatible in this case. We argue that the generalized Gibbs-Duhem identity suggests a revision of the zeroth law which in turns permits to reconsider Maxwell's construction in analogy with the standard case. The physical feasibility of our proposal is considered in the particular case of black holes.