  The large deviations at Level 2.5 are applied to Markov processes with
absorbing states in order to obtain the explicit extinction rate of metastable
quasi-stationary states in terms of their empirical time-averaged density and
of their time-averaged empirical flows over a large time-window $T$. The
standard spectral problem for the slowest relaxation mode can be recovered from
the full optimization of the extinction rate over all these empirical
observables and the equivalence can be understood via the Doob generator of the
process conditioned to survive up to time $T$. The large deviation properties
of any time-additive observable of the Markov trajectory before extinction can
be derived from the Level 2.5 via the decomposition of the time-additive
observable in terms of the empirical density and the empirical flows. This
general formalism is described for continuous-time Markov chains, with
applications to population birth-death model in a stable or in a switching
environment, and for diffusion processes in dimension $d$.
