# Smarr formula for BTZ black holes in general three-dimensional gravity models

## TL;DR

This paper extends the thermodynamic interpretation of BTZ black holes to general three-dimensional gravity models, identifying necessary conditions to preserve the Smarr formula and the first law of thermodynamics.

## Contribution

It demonstrates that direct extension of thermodynamic relations to general 3D gravity models requires additional constraints, refining the understanding of BTZ black hole thermodynamics.

## Key findings

- Extra conditions are needed for the Smarr formula in general 3D gravity models.
- Constraints from thermodynamics align with previous theoretical discussions.
- Thermodynamic interpretation constrains the form of 3D gravity models.

## Abstract

Recent studies have presented the interpretation of thermodynamic enthalpy for the mass of BTZ black holes and the corresponding Smarr formula. All these are made in the background of three-dimensional (3D) general relativity. In this paper, we extend such interpretation into general 3D gravity models. It is found that the direct extension is unfeasible and some extra conditions are required to preserve both the Smarr formula and the first law of black hole thermodynamics. Thus, BTZ black hole thermodynamics enforces some constraints for general 3D gravity models, and these constraints are consistent with all previous discussions.