We develop a novel holographic dictionary for the thermodynamics of black holes with Lifshitz and hyperscaling violating asymptotics, generalizing the dictionary for anti—de Sitter black holes. Using our dictionary we show that the holographic Euler equation is dual to a generalized Smarr formula for these black holes, and we find a precise match between the extended bulk and boundary first law. Notably, the dictionary for the central charge appearing in the Euler relation depends on the hyperscaling violating parameter, but not on the Lifshitz dynamical exponent.
Cong et al. (Tue,) studied this question.