How does the amount of heat energy transferred to cold water depend on the temperature of hot water?

Heat transfer takes place in several ways: conduction (diffusion), convection, and radiation. The water is attempting to reach thermal equilibrium, so the higher the temperatures the quicker the movement toward equilibrium. The second law of thermodynamics states that this heat transfer will take place, moving from higher energy (heat) to lower (cold) until they are both a similar temperature.