  We perform a holographic calculation of the Entanglement R\'enyi entropy
$S_q(\mu,\lambda)$, for spherical entangling surfaces in boundary CFT's with
Einstein-Gauss-Bonnet-Maxwell holographic gravitational duals. We find that for
Gauss-Bonnet couplings $\lambda$, larger than a specific value, but still
allowed by causality, a violation of an inequality that R\'enyi entropies must
obey by definition occurs. This violation is related to the existence of
negative entropy black holes and restricts the coefficient of the Gauss-Bonnet
coupling in the bulk theory. Furthermore, we discover a distinction in the
analytic structure of the analytic continuation of $S_q(\mu,\lambda)$, between
negative and non-negative $\lambda$, suggesting the existence of a phase
transition.
