Can hades leave the underworld as he pleases?

Some myths say he cannot, that he must remain there. But others state that he can, for example when he claimed Persephone, he left the underworld to claim her.

He is also permitted to enter Mt. Olympus, but he chooses not to because the other Gods find him depressing and morbid.

So, in my personal opinion I would say yes, he probably can leave the Underworld as he pleases, but does not have much of a reason to most of the time. Since the Underworld is his Kingdom.