Wall or Window Units
These air conditioners are contained within a single box and are placed in your window or built into a wall. While you may need multiple units to cool a large apartment, small apartments or studios are often cooled to satisfaction with one strong wall or window unit. Keep in mind that running one of these air conditioners nonstop because it isn't adequate for your space will shorten the life of the air conditioner while increasing your energy bill significantly (keep this in mind when looking for a cheap apartment for rent, if price tag is the most important thing for you). Look for energy-efficient units, and try to position window air conditioners in the room in which you will spend most of your time.
Portable Air Conditioners
These air conditioners are designed to move around your apartment, and most come on wheels for convenience. They may not cool more than one room at a time, and you may need to run them continuously while you're at home to maintain a comfortable room temperature. While these units are less energy efficient than the other options, some renters consider them better than nothing.
Ductless Split Systems
These systems are similar to central air units, but they're designed for buildings that don't have ducts. You will have a compressor unit outside of the apartment and a wall unit with a fan inside your apartment. Just as with central heating and air, the impact on your energy bill will depend on the age of the system, how well the system is maintained, and how often the unit runs.
Your final option is to rent an apartment in Tempe without air conditioning. If there are ceiling fans throughout the apartment and you can add a few box fans to circulate air, you may find that this keeps your energy bill low while cooling your environment to satisfaction. You may also ask if you're permitted to bring your own portable or window air conditioner.