If you want to beat the heat as soon as possible go
up not north as north is no longer any guarantee of cool weather.
When we leave Florida we take the quickest route possible to get to I40 at Amarillo. From there on to Flagstaff the climb, cooler temperatures, and lower humidity are steady. Right now we're boondocking at about 8,600 ft near Buford, WY. Cool temps most of the time, 66° right now 53° tonight.
https://www.wunderground.com/weather/us/wy/buford