Gauss-Bonnet black holes in a special anisotropic scaling spacetime

Abstract: Inspired by the Lifshitz gravity as a theory with anisotropic scaling behavior, we suggest a new $(n+1)-$dimensional metric in which the time and spatial coordinates scale anisotropically as $(t,r,\theta_{i})\,\to (\lambda^{z}t,\lambda^{-1}r,\lambda^{x_i}\,\theta_{i})$. Due to the anisotropic scaling dimension of the spatial coordinates, this spacetime does not support the full Schr\"{o}dinger symmetry group. We look for the analytical solution of Gauss-Bonnet gravity in the context of the mentioned geometry. We show that Gauss-Bonnet gravity admits an analytical solution provided that the constants of the theory are properly adjusted. We obtain an exact vacuum solution, independent of the value of the dynamical exponent $z$, which is a black hole solution for the pseudo-hyperbolic horizon structure and a naked singularity for the pseudo-spherical boundary. We also obtain another exact solution of Gauss-Bonnet gravity under certain conditions. After investigating some geometrical properties of the obtained solutions, we consider the thermodynamic properties of these topological black holes and study the stability of the obtained solutions for each geometrical structure.