  We look at the long time behavior of potential Mean Field Games (briefly MFG)
using some standard tools from weak KAM theory. We first show that the
time-dependent minimization problem converges to an ergodic constant
$-\lambda$, then we provide a class of examples where the value of the
stationary MFG minimization problem is strictly greater than $-\lambda$. This
will imply that the trajectories of the time-dependent MFG system do not
converge to static equilibria.
