Varies by apartment, but when I moved into mine, I told them I had geckos and they "checked" with the manager and said that if it lived in a tank / cage it was OK I just had to tell them what I had so that they'd know (i.e. when cleaning or maintenance people went into the apartment). I think they also have a "no breeding" clause, but I think that is geared to cats / dogs (which you're limited to 2 here per apartment).

I'd just look for a place that allows cats / dogs or ask (if they don't) if animals in cages are allowed.