  In this paper we study the scaling of the correction of the Renyi entropy of
the excited states in systems described, in the continuum limit, by a conformal
field theory (CFT). These corrections scale as $L^{-\frac{2\Delta}{n}}$, where
$L$ is the system size and $\Delta$ is the scaling dimension of a relevant bulk
operator located around the singularities of the Riemann surface
$\mathcal{R}_n$. Their name is due to their explicit dependence on the Riemann
surface $\mathcal{R}_n$. Their presence has been detected in several works on
the entanglement entropy in finite size systems, both in the ground and the
excited states. Here, we present a general study of these corrections based on
the perturbation expansion on $\mathcal{R}_n$. Some of the terms in this
expansion are divergent and they will be cured with addition cut-offs. These
cut-offs will determine how these corrections scale with the system size $L$.
Exact numerical computations of the Renyi entropy of the excited states of the
XX model are provided and they confirm our theoretical prediction on the
scaling of corrections. They allow also a comparison with the other works
present in the literature finding that the corrections, for the excited states,
have the exact same form of the ones of the ground state case multiplied by a
model dependend function of $n$ and $l/L$.
