In sunny SoCal, not quite so hot as AZ, I run a big bucket of water in the sink from which I take all the H2O I use for making my Caffenol solution and all the rinses. I add ice or heat up some in the microwave as needed. Doing some stand (mostly semi-stand, I guess), I can't say I have worried about the temperature shift DURING processing-- I imagine it's 'baked in', so to speak, along with the temperature rise from the sodium carbonate, etc. When I'm doing LOW-TEMP stand developing for age fog reduction, I have worried more about it, but again there's only so much I can do with the current facts of my life (aka developing in the kitchen).
I have seen people on this forum speak of basically 'inverting' the temp rise problem , by doing stand dev in the refrigerator. You might be able to try that-- my fridge is too full.