This is not the dominant factor, and this system will still work even if the water is warmer than room temperature. The primary energy sink is the vaporization of water.
The water does not change temperature during that process.
Heating 1 g of water 1C takes 4.184 J. To heat it from from 0C to 25C (about freezing to room temp) it thus takes104.6 J, much less than the 2257 Joules required to vaporize it. These numbers could be modified to properly account for the variace with temp, but the effect actually gets larger then I believe.
No energy leaves the system; it goes into changing the state of water and is stored in the water.
Because I rely on cuda.