What are my rights if my landlord will not fix my A/C?

The Arizona Landlord/Tenant Act makes it clear that a landlord is obligated to maintain air-conditioning in safe working order. 

Failure to Supply Essential Services A.R.S. § 33-1364.  If a landlord fails to provide running water, gas and/or electrical service, or fails to provide reasonable amounts of hot water, heat and/or cooling, then the tenant may give notice to the landlord that he is in breach of the lease. At that point, the tenant has one of the following 3 options:

Option 1: You can hire someone to fix the unit and deduct the cost of the repair from your next month’s rent.  However, the cost of the repair should not be more than 1/2 of your rent. Additioinally, you must use a licensed contractor. You can also purchase a portable room air-condition unit.  Again, the purchase price cannot be more than 1/2 of your rent.

Option 2: You can find substitute housing (e.g. a motel) until the A/C is fixed. If this occurs, you are excused from paying rent for as long as the landlord does not provide the essential service. However, you cannot spend more than 125% of the amount that you pay for daily rent.  This can be deducted from your next month’s rental payment.

Option 3: You can sue for any decrease in the rental value of your property and what you paid in rent, as well as terminate your lease because of your landlord’s failure to comply. 

If you rented a house with air conditioning, then by not fixing it, the landlord is breaching or violating the lease--he is not maintaining the home in the condition or shape it was in when your rented. You could bring a legal action against him to either force him to make payments or to recover damages (e.g. the difference in value between the rent you're paying and what the house without A/C would rent for; the cost of staying in a hotel; etc.). Sometimes you can make the repairs yourself and charge the amount to your landlord or take a credit against rent due, but this can be dangerous--if done improperly, you end up in violation of the lease (for not paying full rent) yourself. You should consult with an attorney to see what your options are and the best way to vinidicate your rights. Good luck.