Put awnings on windows and vigilant use of blackout/insulating blinds and opening closing windows. Arizona gets hot during the daytime but is pretty cool at night and has large temperature swings in day and night. If you close the windows during the day and cover them with insulating shades, then in the evening when it cools down, open the windows, open the blinds and stick a fan in a window and you'll probably decrease your heating and cooling bills significantly.
In Denver (similar large temperature swings) our house stays below 75 all year round without the AC ever turning on, it just takes being active about letting the house breath.
Single pane windows are also probably killing you but an infrared scan of your house by an HVAC/Efficiency guy will tell you better. If you have windows you never open, at least put thin film over the frames to get another layer to stop air movement. You can buy it at a hardware store, $20 should get you the film and 2 sided tape for about 4 windows.
If the house has a "hottest place" it is most likely either a lack of insulation compared to the rest of the house or more or less windows compared to the rest of the house. Since it's an addition the lack of insulation is pretty likely. Look into just insulating that room more. If you have giant windows in that room and it gets too hot, think about blinds.
My house is a little smaller with no kids but before we put solar panels on, we were at $40-60/month in electricity and gas bills.