We study false vacuum decay in scalar field theories in the presence of black holes and a heat bath.
We assume that a black hole is in thermal equilibrium with the heat bath (the Hartle-Hawking state).
Combining analytical and numerical methods, we show that there are no regular Euclidean time-dependent and spherically-symmetric solutions around many types of black holes in this state.
This is in contrast with the flat spacetime where such solutions describe tunneling at low temperature.
We conclude that, for many black hole metrics in various dimensions, the Euclidean solution that is relevant for the decay of the Hartle-Hawking vacuum is always static.