We show, in complete accord with the usual Rindler picture, that detectors with constant acceleration a in de Sitter (dS) and anti-de Sitter (AdS) spaces with cosmological constants measure temperatures , the detector `5-acceleration' in the embedding flat 5-space. For dS, this recovers a known result; in AdS, where is negative, the temperature is well defined down to the critical value , again in accord with the underlying kinematics. The existence of a thermal spectrum is also demonstrated for a variety of candidate wavefunctions in AdS backgrounds.