The goal of this paper is to study the long-time average behavior of the value function of optimal control problems with dynamics associated with a family of vector fields satisfying the Lie Algebra rank condition (Hörmander condition). For such problems the Hamiltonian fails to be coervice in the momentum variable. Nevertheless, by a dynamical approach we prove the existence of a unique critical constant for which the ergodic Hamilton-Jacobi equation admits solutions. Then, we use such a constant to obtain the limit profile of the Cauchy problem as the time horizon goes to infinity.