They're not saying the rate varies but the starting point. Going from 30C to 100C takes less energy than going from 20C to 100C even if specific heat remains physical. Both might get labeled a BTU.
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?".
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.