I tried an experiment with my compost pile two days ago. Humanure pile, sawdust inside cover material, coarse hay outside cover material, pile about 8 months old in an open field area, not shaded. I look at my pile temperature on a regular basis and see a set pattern. Add to the pile one day, two days later notice an increase in temperature, peaks at about day 3, drops on day 5 to its seasonal norm. Lately that norm has been about 115 F but this week on day 5 I added about 5 gallons of cool rainwater to the 115 degree pile and got a 35 degree drop in the thermometer which I assumed would be temporary. Two days later and instead of holding or recovering to 115 the pile went to 122. That is a seven degree increase in temp by adding cool water. Not a huge amount but shows that compost does prefer to be moist. I don't consider mine to normally be dry, sometimes the buckets are sloshy and the present season has not been overly dry. I will experiment further and report back in a few weeks.
Life is too short or my list is too long, not sure which.