  In the present paper, Unruh--DeWitt detectors are used in order to
investigate the issue of temperature associated with a spherically symmetric
dynamical space-times. Firstly, we review the semi-classical tunneling method,
then we introduce the Unruh--DeWitt detector approach. We show that for the
generic static black hole case and the FRW de Sitter case, making use of
peculiar Kodama trajectories, semiclassical and quantum field theoretic
techniques give the same standard and well known thermal interpretation, with
an associated temperature, corrected by appropriate Tolman factors. For a FRW
space-time interpolating de Sitter space with the Einstein--de Sitter universe
(that is a more realistic situation in the frame of $\Lambda$CDM cosmologies),
we show that the detector response splits into a de Sitter contribution plus a
fluctuating term containing no trace of Boltzmann-like factors, but rather
describing the way thermal equilibrium is reached in the late time limit. As a
consequence, and unlike the case of black holes, the identification of the
dynamical surface gravity of a cosmological trapping horizon as an effective
temperature parameter seems lost, at least for our co-moving simplified
detectors. The possibility remains that a detector performing a proper motion
along a Kodama trajectory may register something more, in which case the
horizon surface gravity would be associated more likely to vacuum correlations
than to particle creation.
