# BTZ black hole with KdV-type boundary conditions: Thermodynamics revisited

## TL;DR

This paper explores the thermodynamics of BTZ black holes with KdV-type boundary conditions, revealing anisotropic scaling, a generalized Smarr relation, and a Hawking-Page phase transition at a specific temperature.

## Contribution

It introduces KdV-type boundary conditions for BTZ black holes, deriving an anisotropic Smarr relation and analyzing phase transitions in this new thermodynamic framework.

## Key findings

- Anisotropic Smarr relation consistent with anisotropic Cardy formula
- Generalized thermodynamic ensemble for KdV boundary conditions
- Hawking-Page phase transition at a specific self-dual temperature

## Abstract

The thermodynamic properties of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole endowed with Korteweg-de Vries (KdV)-type boundary conditions are considered. This familiy of boundary conditions for General Relativity on AdS$_{3}$ is labeled by a non-negative integer $n$, and gives rise to a dual theory which possesses anisotropic Lifshitz scaling invariance with dynamical exponent $z=2n+1$. We show that from the scale invariance of the action for stationary and circularly symmetric spacetimes, an anisotropic version of the Smarr relation arises, and we prove that it is totally consistent with the previously reported anisotropic Cardy formula. The set of KdV-type boundary conditions defines an unconventional thermodynamic ensemble, which leads to a generalized description of the thermal stability of the system. Finally, we show that at the self-dual temperature $T_{s}= \frac{1}{2\pi}(\frac{1}{z})^{\frac{z}{z+1}}$, there is a Hawking-Page phase transition between the BTZ black hole and thermal AdS$_{3}$ spacetime.