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We investigate the possibility of having hair on the cosmological horizon. The cosmological horizon shares similar properties of black hole horizons in the aspect of having hair on the horizon. 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 stays 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 Bekenstein's identity method, we also prove the no scalar hair theorem for de Sitter space and de Sitter black holes if the scalar potential is convex.
Cai et al. (Mon,) studied this question.