  We present a large deviation principle for the entropy penalized Mather
problem when the Lagrangian L is generic (in this case the Mather measure $\mu$
is unique and the support of $\mu$ is the Aubry set). Consider, for each value
of $\epsilon $ and h, the entropy penalized Mather problem $\min
\{\int_{\tn\times\rn}
  L(x,v)d\mu(x,v)+\epsilon S[\mu]\},$ where the entropy S is given by
$S[\mu]=\int_{\tn\times\rn}\mu(x,v)\ln\frac{\mu(x,v)}{\int_{\rn}\mu(x,w)dw}dxdv,$
and the minimization is performed over the space of probability densities
$\mu(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E.
Valdinoci that there exists a minimizing measure $\mu_{\epsilon, h}$ which
converges to the Mather measure $\mu$. We show a LDP $\lim_{\epsilon,h\to0}
\epsilon \ln \mu_{\epsilon,h}(A),$ where $A\subset
\mathbb{T}^N\times\mathbb{R}^N$. The deviation function I is given by $I(x,v)=
L(x,v)+\nabla\phi_0(x)(v)-\bar{H}_{0},$ where $\phi_0$ is the unique viscosity
solution for L.
