# Thermodynamic Volume and the Extended Smarr Relation

## TL;DR

This paper investigates the thermodynamic volume and extended Smarr relation for black holes, defining volume as a conjugate to pressure, and deriving an extended Smarr relation without relying on charge scaling.

## Contribution

It introduces a general method to define thermodynamic volume and derive extended Smarr relations using reduced action formalism without Euler scaling assumptions.

## Key findings

- Thermodynamic volume can be explicitly expressed in terms of metric and field variables.
- Extended Smarr relation involving bulk pressure and volume is derived.
- The approach accommodates hairy contributions without difficulty.

## Abstract

We continue to explore the scaling transformation in the reduced action formalism of gravity models. As an extension of our construction, we consider the extended forms of the Smarr relation for various black holes, adopting the cosmological constant as the bulk pressure as in some literatures on black holes. Firstly, by using the quasi-local formalism for charges, we show that, in a general theory of gravity, the volume in the black hole thermodynamics could be defined as the thermodynamic conjugate variable to the bulk pressure in such a way that the first law can be extended consistently. This, so called, thermodynamic volume can be expressed explicitly in terms of the metric and field variables. Then, by using the scaling transformation allowed in the reduced action formulation, we obtain the extended Smarr relation involving the bulk pressure and the thermodynamic volume. In our approach, we do not resort to Euler's homogeneous scaling of charges while incorporating the would-be hairy contribution without any difficulty.