  Consider a topologically transitive unilateral countable Markov shift
$\Sigma$, a locally constant potential $\phi : \Sigma \to \mathbb{R}$
satisfying suitable conditions, and assume that $\mu_t$ is the unique
stationary Markov equilibrium state associated to the potential $t\phi$ for
each $t \geq 1$. In this paper we prove a first level large deviations
principle at zero temperature for the family of equilibrium states $(\mu_t)_{t
\geq 1}$ and we extend the result to the setting of bilateral countable Markov
shifts.
