  It is notoriously difficult to apply statistical mechanics to generally
covariant systems, because the notions of time, energy and equilibrium are
seriously modified in this context. We discuss the conditions under which
weaker versions of these notions can be defined, sufficient for statistical
mechanics. We focus on reparametrization-invariant systems without additional
gauges. The key is to reconstruct statistical mechanics from the ergodic
theorem. We find that a suitable split of the system into two non interacting
components is sufficient for generalizing statistical mechanics. While
equilibrium acquires sense only when the system admits a suitable split into
three weakly interacting components ---roughly: a clock and two systems among
which a generalization of energy is equi-partitioned. This allows the
application of statistical mechanics and thermodynamics as an additivity
condition of such generalized energy.
