Guys, I am gonna side with the OP on this. You are all wrong...provided he is drinking very cold water.
Now, if he drank a gallon of water that was COLD, like 3 degrees Celsius. The body would need to heat that to 34 degrees (human body temp).
It takes 1 calorie to raise a gram of water 1 degree C.
A gallon of water weighs about 3785 grams.
3785 grams x 34 degrees = 128,690 calories.
We are really talking kcals though, so almost 129 calories are burned.
Well, kind of. There is heat in our bodies that is already wasted and just sitting there, ready to heat the water or dissipate. So in the real world this isn't exact...but that gallon could likely save him at least 50 calories.
Drinking 10 gallons of cold cold water is like spending an hour on a treadmill at 5 mph. Shred city.