I'm in AZ too and cooling wort this time of year is tough, especially without an chiller of some kind. GenIke is right, boiling with the full volume of water is ideal. You get better hop utilization, flavor improves, color improves, the list goes on. Do a quick search on here and the reasons will be explained by people more knowledgeable than me.
That being said, I made a lot of delicious beer with partial boils. Plus, when you're just getting into this hobby you generally don't have the equipment that's necessary to do full volume boils (8 gal brew pot, stove or burner that will actually boil that volume, chiller etc.)
I wouldn't recommend dropping below 3 gals on the boil. But what you can do is boil, then add ice directly into the fermenter as a substitute for your top off water. An 8lb bag of ice equals about 1 gallon of water. I did this many times with great results. The wort is chilled in a matter of minutes, instead of hours.
If you want to be extra careful, make sanitized ice yourself by first boiling water then putting it in ice trays and freezing it. Or if that sounds like too much work, find a reputable place that sells ice that's been made from RO water. There's a risk here of contaminating your wort if the ice is dirty, so don't buy any ripped or dirty bags. But I used store bought ice many times with no infection issues. You decide if you're comfortable with the risk.
The great thing about this hobby is that its scalable. Sure more advanced equipment will produce better results. But you can make good beer without it. Be creative and make what you have work until you're ready to take the next step.
