Scalar hair from a derivative coupling of a scalar field to the Einstein tensor

We consider a gravitating system of vanishing cosmological constant consisting of an electromagnetic field and a scalar field coupled to the Einstein tensor. A Reissner-Nordström black hole undergoes a second-order phase transition to a hairy black hole of generally anisotropic hair at a certain critical temperature which we compute. The no-hair theorem is evaded due to the coupling between the scalar field and the Einstein tensor. Within a first-order perturbative approach, we calculate explicitly the properties of a hairy black hole configuration near the critical temperature and show that it is energetically favourable over the corresponding Reissner-Nordström black hole.