H. Narnhofer, I. Peter, W. Thirring How Hot Is the de Sitter Space? (40K, LaTeX) ABSTRACT. We show that the unique invariant locally Minkowskian state of quantum fields in de Sitter space $M$ has for an observer moving along with a Killing vector field a temperature $\frac{1}{2\pi} \sqrt{\frac{1}{R^2} + |a|^2}$ where $R$ is the radius of $M$ and $a$ his acceleration. States with another temperature cannot be locally Minkowskian all over $M$.