In three dimensions, we consider a particular truncation of the Horndeski action that reduces to the Einstein-Hilbert Lagrangian with a cosmological constant Λ and a scalar field whose dynamics is governed by its usual kinetic term together with a nonminimal kinetic coupling. Requiring the radial component of the conserved current to vanish, the solution turns out to be the BTZ black hole geometry with a radial scalar field well defined at the horizon. This means in particular that the stress tensor associated to the matter source behaves on shell as an effective cosmological constant term. We construct a Euclidean action whose field equations are consistent with the original ones and such that the constraint on the radial component of the conserved current also appears as a field equation. With the help of this Euclidean action, we derive the mass and the entropy of the solution, and find that they are proportional to the thermodynamical quantities of the BTZ solution by an overall factor that depends on the cosmological constant. The reality condition and the positivity of the mass impose the cosmological constant to be bounded from above as Λ ≤ − 1 l 2 where the limiting case Λ ¼ − 1 l 2 reduces to the BTZ solution with a vanishing scalar field. Exploiting a scaling symmetry of the reduced action, we also obtain the usual three-dimensional Smarr formula. In the last section, we extend all these results in higher dimensions where the metric turns out to be the Schwarzschild-anti-de Sitter spacetime with planar horizon.

In arbitrary dimension, we consider the Einstein-Maxwell Lagrangian supplemented by the more general quadratic-curvature corrections. For this model, we derive four classes of charged Lifshitz black hole solutions for which the metric function is shown to depend on a unique integration constant. The masses of these solutions are computed using the quasilocal formalism based on the relation established between the off-shell ADT and Noether potentials. Among these four solutions, three of them are interpreted as extremal in the sense that their mass vanishes identically. For the last family of solutions, the quasilocal mass and the electric charge both are shown to depend on the integration constant. Finally, we verify that the first law of thermodynamics holds for each solution and a Smarr formula is also established for the four solutions.

The aim of this letter is to confirm in new concrete examples that the semiclassical entropy of a three-dimensional Lifshitz black hole can be recovered through an anisotropic generalization of the Cardy formula, derived from the growth of the number of states of a boundary non-relativistic field theory. The role of the ground state in the bulk is played by the corresponding Lifshitz soliton obtained by a double Wick rotation. In order to achieve this task, we consider a scalar field nonminimally coupled to new massive gravity for which we study different classes of Lifshitz black holes as well as their respective solitons, including new solutions for a dynamical exponent z = 3. The masses of the black holes and solitons are computed using the quasilocal formulation of conserved charges recently proposed by Gim et al. and based on the off-shell extension of the ADT formalism. We confirm the anisotropic Cardy formula for each of these examples, providing a stronger base for its general validity. Consistently, the first law of thermodynamics together with a Smarr formula are also verified.

In arbitrary dimensions, we consider a particular Horndeski action given by the Einstein-Hilbert Lagrangian with a cosmological constant term, while the source part is described by a real scalar field with its usual kinetic term together with a nonminimal kinetic coupling. In order to evade the no-hair theorem, we look for solutions where the radial component of the conserved current vanishes identically. Under this hypothesis, we prove that this model can not accommodate Lifshitz solutions with a radial scalar field. This problem is finally circumvented by turning on the time dependence of the scalar field, and we obtain a Lifshitz black hole solution with a fixed value of the dynamical exponent z = 1 3. The same metric is also shown to satisfy the field equations arising only from the variation of the matter source.

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.