  We study and look for similarities between the response rates $R^{\rm
dS}(a_0, \Lambda)$ and $R^{\rm SdS}(a_0, \Lambda, M)$ of a static scalar source
with constant proper acceleration $a_0$ interacting with a massless,
conformally coupled Klein-Gordon field in (i) deSitter spacetime, in the
Euclidean vacuum, which describes a thermal flux of radiation emanating from
the deSitter cosmological horizon, and in (ii) Schwarzschild-deSitter
spacetime, in the Gibbons-Hawking vacuum, which describes thermal fluxes of
radiation emanating from both the hole and the cosmological horizons,
respectively, where $\Lambda$ is the cosmological constant and $M$ is the black
hole mass. After performing the field quantization in each of the above
spacetimes, we obtain the response rates at the tree level in terms of an
infinite sum of zero-energy field modes possessing all possible angular
momentum quantum numbers. In the case of deSitter spacetime, this formula is
worked out and a closed, analytical form is obtained. In the case of
Schwarzschild-deSitter spacetime such a closed formula could not be obtained,
and a numerical analysis is performed. We conclude, in particular, that $R^{\rm
dS}(a_0, \Lambda)$ and $R^{\rm SdS}(a_0, \Lambda, M)$ do not coincide in
general, but tend to each other when $\Lambda \to 0$ or $a_0 \to \infty$. Our
results are also contrasted and shown to agree (in the proper limits) with
related ones in the literature.
