On thermodynamics of AdS black holes with scalar hair

It has been known that in the presence of a scalar hair there would be a distinct additional contribution to the first law of black hole thermodynamics. While it has been checked in many examples, a deeper understanding of this issue is necessary. The thermodynamics of AdS black holes in Einstein-scalar gravity is studied by using the standard holographic renormalization procedure and the variation of the Hamiltonian via the Wald formalism. It is found that the first law requires a modification by including an additional term that has a particular form $\sim 〈O〉\delta {\varphi }_s$, with ${\varphi }_s$ and $〈O〉$ a new pair of thermodynamic conjugate variables. ${\varphi }_s$ is the leading source term of the asymptotic fall-off of the scalar field near the AdS boundary, and $〈O〉$ is precisely the response of the dual scalar operator from the holographic point of view. Some hairy black holes are constructed explicitly to check the first law of thermodynamics as well as the thermodynamic relations.