  We prove a no charged scalar hair theorem for static black holes in de-Sitter
spacetime in the region between the event horizon and the cosmological horizon.
The proof does not depend on the assumption of spherical symmetry. It allows
for general non-minimal coupling functions of the scalar field to gravity and
electromagnetic fields, and for higher curvature term corrections to Einstein
gravity. The extension to other asympitotic spacetimes is applicable by
requiring appropriate boundary conditions. Our result excludes the possibility
for spontaneous scalarization of charged scalar around static charged de-Sitter
black holes.
