if you mean you are hostile to the super power then, in my experience, you will not notice much of anything. this 'might' be different if you participate in power play, however i do not.
if you come across a hostile power play npc they may attack on sight.
the mechanics for interacting with power play commanders may be different and i'll leave it someone with more experience to explain how that works, but for interacting with stations and so forth you shouldn't notice any difference in gameplay at all.
note: there is a different level of hostile you can become with a super power, when you amass a large enough bounty from killing clean ships, that is a whole other ballgame. that kind of hostile prevents you from landing (shoot on site upon entering a stations space), constantly harassed by local authority and bounty ships sent after you with extreme prejudice. quite fun actually.