Summary: Consider an \(\alpha\)-Hölder function \(A:\Sigma\to\mathbb R\) and assume that it admits a unique maximizing measure \(\mu_{\max}\). For each \(\beta\), we denote \(\mu_\beta\), the unique equilibrium measure associated to \(\beta A\). We show that \((\mu_\beta)\) satisfies a large deviation principle, that is, for any cylinder \(C\) of \(\Sigma\), \[ \lim_{\beta\to+\infty} \frac1\beta \log\mu_\beta(C)=- \inf_{x\in C}I(x), \] where \[ I(x)= \sum_{n\geq 0} (V\circ \sigma-V-(A-m))\circ \sigma^n(x), \quad m= \int A\,d\mu_{\max}, \] where \(V(x)\) is any strict subaction of \(A\).