Another Arizonian here. I think you'll find the heat + humidity in Missouri and Oklahoma more uncomfortable than the drier heat in New Mexico, Arizona, and California. The largest factor determining temperatures in the Southwest is elevation. Most of Route 66 in New Mexico and Arizona is at fairly high elevations, 5,000 to 7,000 feet, so very high temperatures are not common. I live at 4,600', the temperature exceeds 100° only a couple of days a year. From Kingman, AZ, west, the route drops to lower elevations, so temperatures, especially in California, will be higher.
As mentioned previously, dehydration is more critical than temperature. Relative humidity of 5-10% is common (the renown "dry heat"), so having sufficient water is a necessity. I've ridden most of Rt 66 in Arizona and New Mexico with a 70 oz. Camelbak and one water bottle and never run dry, because places to get water (gas stations, tourist traps, towns, etc.) are probably not more than about 30 miles apart. I can't comment on the California portion; I've not ridden that.
Do take and use plenty of sunscreen. I prefer long-sleeve shirts.
Google "average monthly temperatures" of town along the route to get an idea of what to expect.
Route 66 is a great ride. Take your time and enjoy the many interesting places along the "Mother Road."