And you ought to do some research on mordants. In this case try the experiment on each dye with and without a tablespoon of vinegar in the solution.

Question

maybe if you test water it would probably be the same, good question

AND, you may wish to try more than one color as different colors may react differently.

Does food dye color faster in warm water or cold water?

This looks like a good project to experiment with. You'll need to decide what you want to color -- perhaps several objects -- a piece of cloth, a hard-boiled egg, a piece of celery, and a small piece of wood (like the stick from a taffy apple.)

Then test whether warm or cold water colors these objects faster.