The free energy and entropy for a non-extremal Reissner - Nordström black hole are studied by zeta-function regularization in WKB approximation. Their leading divergences agree with the Susskind - Uglum predictions. Logarithmic singularities appear in connection with the Seeley - De Witt coefficient for the scale anomaly. The equivalence between the small- approximation and the optical space approach is checked. After taking advantage of renormalization group methods, effective potentials are considered for a scalar field with a self-interaction and scalar electrodynamics, both of them on a Schwarzschild background. Numerical study of the potential at the event horizon shows symmetry breaking, and we argue that in some conditions this effect may be a purely quantal one.