BTZ black hole with Korteweg-de Vries-type boundary conditions: Thermodynamics revisited

The thermodynamic properties of the Banãdos-Teitelboim-Zanelli black hole endowed with Korteweg-de Vries (KdV)-type boundary conditions are reconsidered. This family of boundary conditions 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. Finally, we analyze the global and local thermal stability of the system and we found that the specific heat of the black hole is positive for all possible values of z, and at the self-dual temperature Ts=12π(1z)zz+1, there is a Hawking-Page phase transition between the BTZ black hole and thermal AdS3 spacetime.