Question

do hot and cold water take the same time to become normal water at room temperature?

Answer 1

Rate of heating is directly proportional to the temperature difference between liquid and the surrounding.

At the room temperature, temperature difference between the hot water and the room(surrounding) is more, therefore rate of heat transfer is high.
whereas temperature difference between the cold water and the room temperature is less,
rate of heat transfer is less,
therefore hot water take less time to become normal at room temperature than cold water.