Thermodynamics of charged black holes in Einstein-Horndeski-Maxwell theory

We extend an earlier investigation of the thermodynamics of static black holes in an Einstein-Horndeski theory of gravity coupled to a scalar field, by including now an electromagnetic field as well. By studying the two-parameter families of charged static black holes, we obtain much more powerful constraints on the thermodynamics since, unlike in the uncharged one-parameter case, now the right-hand side of the first law is not automatically integrable. In fact, this allows us to demonstrate that there must be an additional contribution in the first law, over and above the usual terms expected for charged black holes. The origin of the extra contribution can be attributed to the behavior of the scalar field on the horizon of the black hole. We carry out the calculations in four dimensions and also in general dimensions. We also derive the ratio of viscosity to entropy for the dual boundary field theory, showing that the usual viscosity bound for isotropic solutions can be violated, with the ratio depending on the mass and charge.