How does the temperature of water hot or cold affect how fast Crayola bathtub tints dissolve?

Basic physics will tell us that higher temperatures will led to a faster rate of diffusion. The water particles will be moving faster (higher temperature = higher kinetic energy = higher particle speed), and this will lead to more particle collisions. With more, collisions, the tints will be spread quicker.

A fun way to visualize this is to get 2 identical drinking glasses. Fill one with hot water, and the other with cold. Then add a single drop of food coloring to each glass at the same time. The coloring will spread quicker in the hot water.