Can an employer fire you without a reason, or notice?

(Texas) For further information, it's a privatly owned business. Got hired as the manager and I'm getting paid 15/h. I've been hearing that HR has been giving interviews for a position in the kitchen unknown if to replace me or not. Can they fire me from one day to the next? I've never done anything wrong. Can they cut my hours or lower my pay? Any information would help.

Thanks for the information guys, hopefully they aren't trying to do what I'm thinking they are.