  In the paper, the temperature associated with a dynamical spherically
symmetric black hole or with a cosmological horizon is investigated from the
point of view of a point-like detector. First, we briefly review the
Hamilton-Jacobi tunneling method for a generic dynamical spherically symmetric
space-time, and present two applications of the tunneling method. Then, we
apply a well-known relativistic quantum theoretical technique, namely the
Unruh-DeWitt detector formalism for a conformally coupled scalar field in a
generic FRW space-time. As an application, for the generic static black hole
case and the FRW de Sitter case, making use of peculiar Kodama observer
trajectories, the tunneling semiclassical results are fully recovered,
automatically corrected by Tolman factors. Some remarks on the temperature of
FRW universe are presented. For more general spaces interpolating de Sitter
space with the Einstein-de Sitter universe a second set of poles is present,
whose exact role remains to be clarified, plus an extra fluctuating term
describing the way equilibrium is reached, similarly to de Sitter space. The
simple thermal interpretation found for de Sitter space is lost and forces, at
a same time, a different quantum interpretation of the horizon surface gravity
for the cosmological FRW models.
