We've been going through an AC fiasco since early July. Moved into a townhouse (in central AL, so yeah, it's hot). The AC ran and blew cool air, but it didn't blow very hard out of the vents downstairs and barely blew at all out of the upstairs vents. Thus the unit ran all the time and the downstairs rarely got below 75, while the upstairs sat at 80+. Our power bill was insanely high because of this. We complained... and complained... and COMPLAINED. We were told by the assistant manager that nobody else was complaining and that if we're hot we should get a window unit.
Funny thing about that is one of the maintenance guys mentioned that multiple people were complaining and having issues. Then later another maintenance guy said he'd lived in one of the units and it was so bad that he had to sleep in the basement to stay cool. Yeah... no issues there!
In any case, the unit stopped blowing cool air, so we were without AC recently for a couple of (miserable) days. FINALLY they replaced the unit. After over two MONTHS of complaining, both to the local office and corporate (in fact, they didn't even really do anything until we went to corporate).
In your case it doesn't sound like you have a corporate office to complain to, but I would keep on your LL. I've never understood why heat is mandatory everywhere but AC is not. Especially in places where it gets very hot.