Towards the Cardy formula for hyperscaling violation black holes

Abstract: The aim of this paper is to propose a generalized Cardy formula in the case of three-dimensional hyperscaling violation black holes. We first note that for the hyperscaling violation metrics, the scaling of the entropy in term of the temperature (defined as the effective spatial dimensionality divided by the dynamical exponent) depends explicitly on the gravity theory. Starting from this observation, we first explore the case of quadratic curvature gravity theory for which we derive four classes of asymptotically hyperscaling violation black holes. For each solution, we compute their masses as well as those of their soliton counterparts obtained through a double Wick rotation. Assuming that the partition function has a certain invariance involving the effective spatial dimensionality, a generalized Cardy formula is derived. This latter is shown to correctly reproduce the entropy where the ground state is identified with the soliton. Comparing our formula with the one derived in the standard Einstein gravity case with source, we stress the role played by the effective spatial dimensionality. From this observation, we speculate the general form of the Cardy formula in the case of hyperscaling violation metric for an arbitrary value of the effective spatial dimensionality. Finally, we test the viability of this formula in the case of cubic gravity theory.