I remember last summer on a particularly hot and muggy night we had the air con running inside, keeping the house a pleasant 25°C. I took the scope outside, thinking it would be good to have the mirror well cooled before going out.
Well it took less than 3 minutes for the mirror to get that frosty wet look on it, the same that you get on a glass full of coke and ice. Another 3 minutes and it was wringing wet, water droplets running of it like there was a rainstorm inside the tube.
Yes, it's good to have a mirror cool to start - no, not noticably cooler than outside ambient temp especially when it's quite humid outside. I think ti took the mirror longer to stabilise it's temp than it does normally.
