Black Hole Entropy and Viscosity Bound in Horndeski Gravity

Horndeski gravities are theories of gravity coupled to a scalar field, in which the action contains an additional non-minimal quadratic coupling of the scalar, through its first derivative, to the Einstein tensor or the analogous higher-derivative tensors coming from the variation of Gauss-Bonnet or Lovelock terms. In this paper we study the thermodynamics of the static black hole solutions in $n$ dimensions, in the simplest case of a Horndeski coupling to the Einstein tensor. We apply the Wald formalism to calculate the entropy of the black holes, and show that there is an additional contribution over and above those that come from the standard Wald entropy formula. The extra contribution can be attributed to unusual features in the behaviour of the scalar field. We also show that a conventional regularisation to calculate the Euclidean action leads to an expression for the entropy that disagrees with the Wald results. This seems likely to be due to ambiguities in the subtraction procedure. We also calculate the viscosity in the dual CFT, and show that the viscosity/entropy ratio can violate the $\eta/S\ge 1/(4\pi)$ bound for appropriate choices of the parameters.