I bow hunt almost exclusively from trees stands, most of which are climbing stands (also use a ladder stand on one site). I've found that a fanny pack is a must. I keep my calls, light, phone, etc in there. To keep it out of your way when shooting, just twist it around on your waist so that it's on your off-hand side.
As for keeping cool, I'll take my outer jacket or shirt off on the way to the stand location so that I don't overheat. Once I reach my stand location, I'll set up the stand, tie my haul line on my bow, etc, and only then will I put on all of my clothing. If you're proficient at climbing and don't rush, you can usually make the climb without getting too overheated. Keep your headgear off while you climb, both to help keep cool as well as for safety (easier to see without a mask). Once in the tree, I'll wait about 5 minutes to cool off before donning mask and hat, etc and hauling up my bow.
You'll find a routine that works best for you...after a lot of trial and error, I've settled upon something that works well for me and is pretty fast, efficient, and quiet. You'll figure it out after you do it a few times.