Warped AdS$_3$ black hole thermodynamics and the charged Cardy formula

We revisit the thermodynamic properties of black holes with warped AdS$_3$ asymptotics in topologically massive gravity. The holographically dual theory, often referred to as warped CFT$_2$, exhibits a single copy of the Virasoro-Kac-Moody algebra. Consequently, the asymptotic density of states in the right-moving sector is described by the charged Cardy formula that we review in detail. By defining specific linear combinations of the gravitational thermodynamic potentials, we are able to present the gravitational left- and right-moving on-shell actions directly in the grand-canonical ensemble. This allows us to demonstrate that, apart from the charged Cardy behavior in the right-moving sector, the dual field theory exhibits an additional frozen left-moving state giving rise to non-zero entropy contribution. Our findings elucidate the nature of warped CFTs and reveal subtle yet fundamental differences from the existing literature.