Does anybody roughly now the basics of why you can have hairy black holes in more than 4 D?
You can have hairy black holes, that's for sure. The question is that I don't know why. :(
If you don't no why then you can't be sure can you? If someone says you can have hairy black holes why take their word for it?
I do recall that there are more black hole solutions in higher dimensions...ring solutions
so i believe this paper contains the answers you are looking for http://arxiv.org/abs/hep-th/0608012