We construct examples of Tonelli Hamiltonians on $\T^n$ (for any $n\geq 2$)
such that the hypersurfaces corresponding to the Ma\~n\'e critical value are
stable (i.e. geodesible). We also provide a criterion for instability in terms
of closed orbits in free homotopy classes and we show that any stable energy
level of a Tonelli Hamiltonian must contain a closed orbit.