Thermodynamics of Higher Spin Black Holes in AdS$_3$
We discuss the thermodynamics of recently constructed three-dimensional higher spin black holes in $SL(N,\mathdsR)× SL(N,\mathdsR)$ Chern-Simons theory with generalized asymptotically-anti-de Sitter boundary conditions. From a holographic perspective, these bulk theories are dual to two-dimensional CFTs with $\mathcalW_N$ symmetry algebras, and the black hole solutions are dual to thermal states with higher spin chemical potentials and charges turned on. Because the notion of horizon area is not gauge-invariant in the higher spin theory, the traditional approaches to the computation of black hole entropy must be reconsidered. One possibility, explored in the recent literature, involves demanding the existence of a partition function in the CFT, and consistency with the first law of thermodynamics. This approach is not free from ambiguities, however, and in particular different definitions of energy result in different expressions for the entropy. In the present work we show that there are natural definitions of the thermodynamically conjugate variables that follow from careful examination of the variational principle, and moreover agree with those obtained via canonical methods. Building on this intuition, we derive general expressions for the black hole energy and entropy which are written entirely in terms of the Chern-Simons connections, and are valid for both static and rotating solutions. We compare our results to other proposals in the literature, and provide a new and efficient way to determine the generalization of the Cardy formula to a situation with higher spin charges.