Sub-Riemannian systems are an important class of nonlinear control systems with linear dependence on controls. Controllability properties for such systems are derived by the so-called Lie Algebra rank condition on the associated family of vector fields. We will discuss the long-time average behaviour of the value function of optimal control problems for sub-Riemannian systems, which cannot be addressed by classical weak KAM theory as the Hamiltonian fails to be coercive in the momentum variable.