[stavros]

They specifically said "one degree Fahrenheit", so the GP's question was "why does going from 30 F to 31 F take a different amount of energy than going from 20 F to 21 F?".

[ChiperSoft]

Well in the later scenario the water would be frozen and conduct heat within the volume poorly, so applying heat would result in a phase change at the contact point. Latent heat transfer is then going to complicate things.

[stavros]

It doesn't matter how well it conducts the heat, since you measure the amount the energy you apply to it (after losses).