Regional discussion and conditions reports for the great state of Utah, from the alpine peaks to the desert slots. Please post partners requests and trip plans here or in the Utah Climbing Partners section.
Don't know anything about conditions, but in terms of heat, I'd say beginning of June is when temps begin to get pretty hot at the base and the area surrounding the mountain. The upper parts of the mountain should be pretty comfortable all throughout the summer though.
Expect some snow way up top, but the road should be walkable. Your main issue will be how far up the road you can drive. It gets steep and rough at about 7,200 feet. Assuming you park around there, you're looking at 9 miles r/t, 3,000+ ft of gain, so it's not too bad. It may not get bladed early in the season, so rocks and some deadfall may be an issue.
I went in July, with temps at the base around 100 degrees during the day. I parked at around 7,200 feet. However, I started walking at 0-dark-30, which meant I summitted at around 6:30 am and was down before 10. It was toasty as I returned to my truck. It's a beautiful place. I hope you have a lot of fun!