How hot de Sitter space and black holes can be?

We study quantum fields on spacetimes having a bifurcate Killing horizon by allowing the possibility that left-- and right-- (in--going and out--going) modes have different temperatures. We consider in particular the Rindler for both massless and massive fields, the static de Sitter and Schwarzschild black hole backgrounds for massive fields. We find that in all three cases, when any of the temperatures is different from the canonical one (Unruh, Hawking and Gibbons--Hawking, correspondingly) the correlation functions have extra singularities at the horizon.