  We study the gravitational field equations in the presence of a coupling
between the derivative of a massless scalar field and the Einstein tensor. This
configuration is motivated by Galileon gravity as it preserves shift invariance
in the scalar sector. We analytically obtain solutions with static and
spherically symmetric geometry, which also include black holes with a single
regular horizon. We examine the thermodynamical properties of these solutions,
and we reveal the non-perturbative nature of the Galileon coupling constant. We
also find a phase transition, similar to the one described by Hawking and Page,
which occurs at a critical temperature determined by both the black hole mass
and by the strength of the coupling.
