AC is not a requirement for habitability, unlike heat, electricity and hot water. AC is considered an amenity. That said, a landlord can be required to repair or replace as necessary. If the property was leased to you with central air conditioning or window units and the rental agreement does not specifically exclude them from landlord responsibility, the landlord is responsible. However, a rental agreement may exclude AC, and no state law will override that if it's in the contract.
If AC is the landlord's responsibility, good luck fining a written statute for the required time to make repairs, with 50 states and many, many more local jurisdictions within states. It is generally considered that landlords must make requested repairs within a 'reasonable' period of time. What is reasonable? That's up to the individual jurisdiction and in the end, when someone has to decide what's reasonable, it's a judge in a legal proceeding.
I found one reference to California, and it is not official state law, it's from a San Francisco newspaper, that says 14 days to one month is 'reasonable' for a landlord to make repairs after they have been notified in writing. You always make notification in writing, keeping copies and getting proof of delivery. This paper suggests the 'repair and deduct' remedy - fix it yourself and deduct the cost from rent - but this is something that may not work in every jurisdiction. How to force a landlord to make repairs and what you can legally do if they don't are covered somewhere in state and local landlord-tenant law, but that's up to you to research.