The air conditioning failed. Does my landlord policy cover it?
Not under the standard form. A mechanical failure is not a covered peril, unlike storm or fire damage to the same unit. Equipment breakdown cover is a separate endorsement that addresses exactly this, and on an Arizona rental it is worth having.
How fast do I have to fix it?
Quickly. Cooling is a habitability matter in Arizona summers and the obligation is real. That is the practical reason to carry equipment breakdown rather than planning to absorb it, because the repair cannot wait for a convenient month.
Does it cover the appliances too?
Usually, where they are owned by you and fail mechanically or electrically. It also typically covers resulting damage, such as water from a failed water heater, which is often the larger part of the loss.