  We extend all known black hole no-hair theorems to space-times endowed with a
positive cosmological constant $\Lambda.$ Specifically, we prove that static
spherical black holes with $\Lambda>0$ cannot support scalar fields in convex
potentials and Proca-massive vector fields in the region between black hole and
cosmic horizons. We also demonstrate the existence of at least one type of
quantum hair, and of exactly one charged solution for the Abelian Higgs model.
Our method of proof can be applied to investigate other types of quantum or
topological hair on black holes in the presence of a positive $\Lambda.$
