Definitely cotton to stay cool because it will retain moisture much longer for evaporative cooling. It will also retain your sweat which will help. Soak your whole body at every oportunity. If you can submerge yourself completely, do it and stay there as long as you can stand it. It will take the heat from your core. I wear shorts and short sleeves with a wide-brimmed straw hat and cotton bandana but I wear shorts every day in the warm months, and my legs can take the sun. I soak everything including the hat at every water stop or creek crossing. I drank 4-5 liters of water on my hike in last year, in early Sep. I live and work in the heat so I am more adjusted to it than most.

I probably don't have to tell you, but July is the hottest month and it will be brutal, especially if you enter the inner gorge. Not sure what your every day lifstyle is like, but you should do some hot weather conditiong before you go. Are you hiking the corridor trails? If so stay away from S Kaibab. There is not an inch of shade or water.