False vacuum decay is a key feature in quantum field theories and exhibits a
distinct signature in the early Universe cosmology. It has recently been
suggested that the false vacuum decay is catalyzed by a black hole (BH), which
might cause the catastrophe of the Standard Model Higgs vacuum if primordial
BHs are formed in the early Universe. We investigate vacuum phase transition of
a scalar field around a radiating BH with taking into account the effect of
Hawking radiation. We find that the vacuum decay rate slightly decreases in the
presence of the thermal effect since the scalar potential is stabilized near
the horizon. However, the stabilization effect becomes weak at the points
sufficiently far from the horizon. Consequently, we find that the decay rate is
not significantly changed unless the effective coupling constant of the scalar
field to the radiation is extremely large. This implies that the change of the
potential from the Hawking radiation does not help prevent the Standard Model
Higgs vacuum decay catalyzed by a BH.