  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 elec- tromagnetic 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 behaviour 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.
