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.