  Recently it was shown that there is an anomalous singularity of propagators
in spacetimes with horizons for thermal states with non--canonical
temperatures. In this paper we extend these observations to the situation when
the background geometry is given by the four--dimensional Schwarzschild and
Reissner-Nordstr\"om black holes. Namely, we demonstrate that the two-point
function in the free scalar field theory acquires anomalous singularity when
the two points are located on the horizon. This singularity is anomalous in the
sense that the coefficient before the divergent term in the two-point function
differs from the canonical one and depends explicitly on the temperature of the
state. As it was previously shown, it leads to the explosive behavior of the
regularised stress-energy tensor on the horizon when the temperature of the
state does not coincide with the canonical temperature of the horizon.
