I bet that would actually be kinda nice in winter, lol!
Water cools as soon as it hits the tank, unless you flush it a bunch of times in a row you would never know. Only reason I realised my mistake was that the valve leaked and it was hot after a couple flushes... smh
Took about 5 minutes to fix when I realized what I did. Just tapped the wrong line during the remodel. But it was a DIY screw up of mega proportions for me.
BTW, this was in my very early twenties.