Planar black holes configurations and shear viscosity in arbitrary dimensions with shift and reflection symmetric scalar-tensor theories

In higher dimensions, we explore planar hairy black hole configurations for a special subclass of the Horndeski theory, defined by two coupling functions depending on the kinetic term and enjoying shift symmetry and reflection symmetry. For this analysis, we derive a set of new solutions, given by time-dependent as well as time independent scalar field configurations. Additionally, we calculate their thermodynamic quantities by using Wald formalism, satisfying the First Law of Thermodynamics as well as a Smarr relation. Together with the above, the Wald procedure allows us to compute the shear viscosity, showing that for a suitable choice of the coupling functions the Kovtun-Son-Starinets bound is violated.