  Previous work on dynamical black hole instability is further elucidated
within the Hamilton-Jacobi method for horizon tunneling and the reconstruction
of the classical action by means of the null-expansion method. Everything is
based on two natural requirements, namely that the tunneling rate is an
observable and therefore it must be based on invariantly defined quantities,
and that coordinate systems which do not cover the horizon should not be
admitted. These simple observations can help to clarify some ambiguities, like
the doubling of the temperature occurring in the static case when using
singular coordinates, and the role, if any, of the temporal contribution of the
action to the emission rate. The formalism is also applied to FRW cosmological
models, where it is observed that it predicts the positivity of the temperature
naturally, without further assumptions on the sign of the energy.
