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.
