One rifle / cartridge for AZ hunting... HMMM... If I'm limited to what I own it would probably be my Ruger 77 MkII in .300wmg. Why? Because when I think about hunting big game in AZ it starts with Elk and the other options in my safe today don't buy me enough of anything to make the cartridge downgrade from .300wmg worth it. That cartridge can let you shoot antelope at 500m+ it is plenty for elk, you can puppy load it and take deer at "normal" ranges without jellying 1/2 the animal. I'd have no concern hunting bear, cats, or any other AZ mammal with it if I needed to. It's certainly not "optimal" for smaller stuff - but again - you can load to the specifics of the need. So, it "can" do it all - with only two down-sides... recoil and wasted meat if you shoot smaller critters with hot loads.
If I were buying only one NEW gun for hunting in AZ I'd probably go with a quality bolt gun in 7mm rem mag or something with a similar bullet diameter and long legs - maybe even one of the 6.5s if I was committed to reloading for it. Again - "enough" cartridge for elk, flat enough for antelope out on the planes, and easier to load "down" for deer etc.
Neither of these are my "you can only have one gun for the zombie apocalypse" gun... that is the M1A in 7.62X51 with many many magazines and good glass.