  We calculate the R\'enyi entropy $S_q(\mu,\lambda)$, for a spherical
entangling surface in CFT's with Einstein-Gauss-Bonnet-Maxwell holographic
duals. R\'enyi entropies must obey some interesting inequalities by definition.
However, for Gauss-Bonnet couplings $\lambda$, larger than a specific value,
but still allowed by causality, we observe a violation of the inequality
$\frac{\partial}{{\partial q}}\left({\frac{{q - 1}}{q}S_q(\mu,\lambda)} \right)
\ge 0$, which is related to the existence of negative entropy black holes,
providing interesting restrictions in the bulk theory. Moreover, we find an
interesting distinction of the behaviour of the analytic continuation of
$S_q(\mu,\lambda)$ for imaginary chemical potential, between negative and
non-negative $\lambda$.
