De Sitter radiation as discussed by Davies in this paper is an instance of the Unruh effect in curved spacetime, as I discussed, e.g., in this answer.
As far as I know, there is no direct experimental evidence of the detection of this sort of radiation. However, there is indirect evidence in analogous situations. Namely
- For the Unruh effect in flat spacetime, the existence of the effect can be understood as a requirement for consistency with usual Quantum Field Theory in flat spacetime, as discussed in arXiv:gr-qc/0205078
- For the Unruh effect in flat spacetime, the existence of the effect can be understood as a prediction of classical electrodynamics. This is discussed in arXiv:1701.03446 [gr-qc]
- arXiv:1903.00043 [gr-qc] "present[s] the evidence for the first observation of acceleration-induced thermality in a non-analogue system." (quoted from its abstract). Also, see arXiv:2205.06591 [gr-qc]
- For Hawking radiation, an effect analogue to the Unruh effect, there is direct observation in analogue systems, as discussed by DOI:10.1038/s41586-019-1241-0
In short, there is no direct evidence of de Sitter radiation. Nevertheless, there is indirect evidence in similar situations.