  We consider an overdamped Brownian particle subject to an asymptotically flat
potential with a trap of depth $U_0$ around the origin. When the temperature is
small compared to the trap depth ($\xi=k_B T/U_0 \ll 1$), there exists a range
of timescales over which physical observables remain practically constant. This
range can be very long, of the order of the Arrhenius factor ${\rm e}^{1/\xi}$.
For these quasi-equilibrium states, the usual Boltzmann-Gibbs recipe does not
work, since the partition function is divergent due to the flatness of the
potential at long distances. However, we show that the standard Boltzmann-Gibbs
(BG) statistical framework and thermodynamic relations can still be applied
through proper regularization. This can be a valuable tool for the analysis of
metastability in the non-confining potential fields that characterize a vast
number of systems.
