[kgwgk]

The question was

> Why would the amount of energy required to increase the temperature of a defined mass of water depend on the temperature?

and the answer is that it does, except if you have a very restrictive model where it doesn't. (Edit: I see there was an edit to the previous comment, I had missed that.)

[bonsai_spool]

Yes, I wrote both comments.

Why does water relate to the ideal gas law?

[kgwgk]

It doesn't. Your initial assumption that the heat capacity is a thing that doesn't depend on temperature did (because it may apply to ideal gases as opposed to water where it just doesn't stay constant as you noticed later).