Having been through a similar situation myself this is what I can tell you. Unlike a heater, and air conditioner is not "necessary" for living. I live in the arm-pit of California and it gets HOT here, too. I think it is a necessity, but apparently the courts, etc. do not. You can't withhold your rent even if they don't fix it. You can TRY to send them a certified letter requesting they fix the AC in a reasonable amount of time, this way you have PROOF of asking...but, if they don't do it, I don't think you have any recourse.
The first summer we moved into our apartment, we ran the AC 24/7 and when it was 105 outside it was 98 inside. The AC did nothing. Not to mention, our FIRST MONTH'S electric bill was $336, I kid you not. They would not replace the unit, saying it has worked for 30 years, so there is no reason
. Whatever...there was nothing we could do. We have since put in an evaporative cooler (OUR cooler...we paid for it...it is a window unit) to help somewhat, but it still gets really hot in here.
Good luck to you...I hope you didn't end up with a slum lord like we did! :angryfire