There is nothing about desert conditions that makes carrying a firearm any more or less useful than many other locales. Proximity to our southern border might be another matter, especially if you are getting into secluded areas, which doesn't sound likely, since you are evidently staying on pavement. I haven't spent any time near the border recently, but I have lots of experience with sunny southern Arizona....

On a hot day in the desert, water is everything. The rest of your gear is relatively insignificant. "Ration your sweat, not your water" is a common motto. Don't exert yourself in mid day heat, but slow down and take a siesta. Wait for the cool of the evening and early morning to exert yourself or do any significant walking.

Note that construction workers and other outdoor types get early morning starts - typically with first light. Plan your schedule for early starts and get off the roads and out of the sun as early as possible.

A full floppy brimmed hat is a very important item, preferably of cotton (so you can soak it).

Finally, July is the beginning of "monsoon season" in southern Arizona and it is possible that you may have all the water you can handle. Flooding is a real possibility, so do not stay in dry streambeds and prepare to deal with thunderstorms and (briefly) heavy rains.

You were somewhat vague about your precise location, but check the elevations you will be traveling. If you get over 4000 feet ASL, the day may be hot, but you won't be in desert country. I am leaving for South Dakota to dig mammoths in a few days, and day time temps exceeding 105 degrees will be routine, but no one regards the Black Hills as "desert." This time of year, you need to be prepped for hot weather nearly everywhere in the continental US (except coastal California, where I now reside....)