In order to study physical effects of quantum fields on curved  spacetimes, one needs appropriate, ideally physically well-motivated  Hadamard states to describe the fields. One example of such a state on  black-hole spacetimes is the Unruh state. In this talk, we present a  rigorous construction, including the proof of the Hadamard property, of its generalization to the free  scalar field on slowly rotating Kerr-de Sitter spacetimes.