I mist the cage so much that I actually worry about the plants getting a chance to dry out. I almost never have to water plants in cages.
I have watered plants with well water which tends to be pretty cold with no ill effects.
it's not so cold that my chams run away from it. it's really not that big of a deal. i have a mistking so my water reservoir is room temp. i don't spray w/ a bottle so if i wanted to spray warm water, i would have to install an inline water heater. keeping a submerged water heater in the reservoir is not recommended as it can cause and grow bacteria in the water reservoir.
do you use an automated mister like a mistking or aquazamp or a hand sprayer. even when i use a hand sprayer for when i take my chams outside, i just use room temp water.