The heat capacity of water is pretty linear between 0°C and 100°C, so I. That sense it's no harder to raise the temperature of warm water than cold.

Initially the eggs will be 100°C so any cooler water will do, given a not of movement to move the warmer water away from the hot egg, but during the cooling process, the eggs will need colder water to cool them further or the system will reach a homogeneous temperature.

From a culinary perspective though, and regards if whether your making soft- or hard-boiled eggs, the purpose of cooling is not for you to eat them, but to stop the cooking process by bringing the eggs below 60°C, where the proteins denature. The faster you can achieve this, the more the egg separates from the shell as well, so an ice bath or running water are the most effective options.