  Quantum dynamics (i.e., the Schr\"odinger equation) and classical dynamics
(i.e., Hamilton equations) can both be formulated in equal geometric terms: a
Poisson bracket defined on a manifold. In this paper we first show that the
hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can
also be formulated within this general framework, what has been used in the
literature to construct propagation schemes for Ehrenfest dynamics. Then, the
existence of a well defined Poisson bracket allows to arrive to a Liouville
equation for a statistical ensemble of Ehrenfest systems. The study of a
generic toy model shows that the evolution produced by Ehrenfest dynamics is
ergodic and therefore the only constants of motion are functions of the
Hamiltonian. The emergence of the canonical ensemble characterized by the
Boltzmann distribution follows after an appropriate application of the
principle of equal a priori probabilities to this case. Once we know the
canonical distribution of a Ehrenfest system, it is straightforward to extend
the formalism of Nos\'e (invented to do constant temperature Molecular Dynamics
by a non-stochastic method) to our Ehrenfest formalism. This work also provides
the basis for extending stochastic methods to Ehrenfest dynamics.
