Something that has long puzzled me is the use of full mail hauberks and the like in the Crusades. How did Crusaders and their various enemies avoid cooking inside their armor? I know many of them did die of exposure and the like but they did still fight and take territory. I'd assume the Byzantines and other local powers who used Cataphracts and similarly "covered" troops, ran into similar problems. Was it just a man it up and take it kind of a thing or did they fight at less hot times of the day? Change the campaign season?
I don't have much historical evidence to bring to this one, but I've worn heavy SCA armour on hot days (hot by British Isles standards) and discussed the problem with people who have done so in hotter climates (Texas and Israel, most notably).
So, first and foremost, they probably would not have worn the armour unless they were expecting to go into a fight. If they were travelling from one place to another, you might possibly wear the helm and a breastplate, but not much more. They'd only armour up fully before a battle. The concepts of today's guerilla warfare didn't really apply; many of the Crusader battles were sieges, one way or another, so they were pretty predictable.
Second, chain mail is heavy, but it's not airtight. Some breeze gets through, and with a cotton or even light woolen gambeson below that can be soaked, it's not that uncomfortable. It's certainly tiring, though. I don't imagine that many of the Crusaders would have worn full plate, which would indeed have the 'cooking' effect - full plate was really only coming in as the Crusades were coming to an end.
In any case, only a few of the people on the Crusades would have actually worn much armour - the nobles and knights. Their men-at-arms, infantrymen, and just plain foot soldiers would have been lucky to have a helm over a padded jack, rather than anything heavier.
After that - they'd have fought in cooler weather whenever possible, kept hydrated as much as possible (not that they had the term, but they certainly had the concept), and most importantly, not actually got into that many battles. A successful (that is, surviving) Crusader might have been in only a couple of real battles, and maybe a few scuffles on the side, many without armour.
I'm not sure but in the Encyclopaedia World History book it said many of them fought without armour, while others soaked cloth in water and put it underneath their armour. I would guess that under such heat exposure, as you said before, many died. Some of the battles, like you said, took place at night or less hot parts of the day. The armour they wore might have been heavy but may have still provided shade from the sun in some cases. Many coated themselves in oil; this may be because the oil could potentially repel the heat, but I'm not so sure about that one.