# Hairy Black Hole Chemistry

## TL;DR

This paper explores the thermodynamics of hairy black holes in AdS spacetime, revealing unique critical phenomena due to scalar fields and extending the analysis to include a dynamical cosmological constant.

## Contribution

It introduces a detailed thermodynamic analysis of hairy black holes with scalar fields, including boundary term methods and criticality in extended phase space.

## Key findings

- Existence of two critical points in the canonical ensemble.
- Scalar field self-interaction enables criticality in grand canonical ensemble.
- Verification of the isoperimetric inequality for hairy black holes.

## Abstract

We study the thermodynamics of an exact hairy black hole solution in Anti-deSitter (AdS) spacetime. We use the counterterm method supplemented with boundary terms for the scalar field to obtain the thermodynamic quantities and stress tensor of the dual field theory. We then extend our analysis by considering a dynamical cosmological constant and verify the isoperimetric inequality. Unlike the thermodynamics of Reissner-N\"{o}rdstrom (RN) black hole in this `extended' framework, the presence of the scalar field and its self-interaction make also the criticality possible in the grand canonical ensemble. In the canonical ensemble, we prove that, in fact, there exist two critical points. Finally we comment on a different possible interpretation that is more natural in the context of string theory.