Uh, no, its a lot more simple than that, and can be illustrated with a question: has your urine ever been cold? I'm betting that the answer is "no" and that's because the body brings all food and drink you consume to body temperature. 1 1/2 gallons of very cold water would have to be heated up from maybe 40 degrees to 98.6 - and that takes a lot of energy. THAT is why you burn more calories over time.
By my calculation, raising 1.5 gallons of water 60 degrees F uses about 190 Calories.
That's not huge, but I suppose over time it would add up--provided you did not change the amount you eat to compensate.
Anybody want to check my arithmetic?
1 Calorie = 1000 calories
1 degree C = 1.8 degrees F
1 gallon (water) = 8.33 lb
1 lb = 454 gm