  In this paper, we study the scalar field condensation around the hyperbolic
black hole solutions in the Einstein and Gauss-Bonnet gravities. We investigate
the Renyi entropy and inequalities governing on it under this phase transition.
Our numerical computations show that for the positive values of the
Gauss-Bonnet coupling and below a critical temperature one of these
inequalities is violated. This puts more restrictions on the allowed values of
the Gauss-Bonnet coupling.
