Right sorry, we know that P != EXP from the hierarchy results. (Gah. Should have realize that made no sense to think that P != PSPACE could come from from hierarchy results, given that one is a time defined set and one is a space defined set in order to get that one would probably need an intermediate computational class or a deep equivalence).

Edit: I should also probably specify that Sniffnoy was the person who made me aware of the the Aaronson work cited above.