  We investigate the possibility of having hairs on the cosmological horizon.
The cosmological horizon shares similar properties of black hole horizons in
the aspect of having hairs on the horizons. For those theories admitting haired
black hole solutions, the nontrivial matter fields may reach and extend beyond
the cosmological horizon. For Q-stars and boson stars, the matter fields cannot
reach the cosmological horizon. The no short hair conjecture keeps valid,
despite the asymptotic behavior (de Sitter or anti-de Sitter) of black hole
solutions. We prove the no scalar hair theorem for anti-de Sitter black holes.
Using the Bekenstein's identity method, we also prove the no scalar hair
theorem for the de Sitter space and de Sitter black holes if the scalar
potential is convex.
