  Let $\Sigma_{A}(\mathbb{N})$ be a topologically mixing countable Markov shift
with the BIP property over the alphabet $\mathbb{N}$ and $f:
\Sigma_{A}(\mathbb{N}) \rightarrow \mathbb{R}$ a potential satisfying the
Walters condition with finite Gurevich pressure. Under suitable hypotheses, we
prove the existence of a Large Deviation Principle for the family
$(\mu_{\beta})_{\beta > 0}$ where each $\mu_{\beta}$ is the Gibbs measure
associated to the potential $\beta f$. Our main theorem generalizes from finite
to countable alphabets and also to a larger class of potentials a previous
result of A. Baraviera, A. O. Lopes and P. Thieullen.
