  Using the Schwarzschild metric as a rudimentary toy model, we pedagogically
revisit the curious prediction that the mass of a classical black hole in a
constant temperature thermal bath diverges in a finite amount of time. We study
in detail how this instability behaves if the temperature of the bath is
allowed to vary with time and conclude that whatever the background behavior
(but for a zero-measure subspace of the initial conditions), the black hole
mass either diverges or vanishes in a finite time if the Hawking radiation is
taken into account. The competition between both effects is subtle and not
entirely governed by the hierarchy of the relevant temperatures. This
instability is also shown to be reached before the background singularity in a
contracting universe, which has implications for bouncing models. The results
are generalized to spaces with extra dimensions and the main conclusions are
shown to remain true. The limitations of the model are reviewed, both from the
point of view of the dynamical black hole horizon and from the point of view of
the background space expansion. Comparisons with other approaches are suggested
and possible developments are underlined.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Astrophysical Phenomena;High Energy Physics - Phenomenology