In flat space and at finite temperature, there are two regimes of false
vacuum decay in quantum field theory. At low temperature, the decay proceeds
through thermally-assisted tunneling described by periodic Euclidean solutions
— bounces — with non-trivial time dependence. On the other hand, at high
temperature the bounces are time-independent and describe thermal jumps of the
field over the potential barrier. We argue that only solutions of the second
type are relevant for false vacuum decay catalyzed by a black hole in
equilibrium with thermal bath. The argument applies to a wide class of
spherical black holes, including $d$-dimensional AdS/dS-Schwarzschild black
holes and Reissner-Nordstr\”om black holes sufficiently far from criticality.
It does not rely on the thin-wall approximation and applies to multi-field
scalar theories.
