  Recently, in [Class. Quantum Grav. 33 (2016) 045005], the authors proposed a
new approach extending the framework of statistical mechanics to
reparametrization-invariant systems with no additional gauges. In this work,
the approach is generalized to systems defined by more than one Hamiltonian
constraints (multi-fingered time). We show how well known features as the
Ehrenfest- Tolman effect and the J\"uttner distribution for the relativistic
gas can be consistently recovered from a covariant approach in the
multi-fingered framework. Eventually, the crucial role played by the
interaction in the definition of a global notion of equilibrium is discussed.
