What is the ideal nutrient water temperature? Should it be cooler than the grow room?

The roots are especially sensitive to low temperatures. The absorption of nutrients through the roots is an active process. This means that the root needs energy to absorb the nutrients, but also to be able to select these nutrients. The root can, to a certain point, choose which and how much nutrient it absorbs. This process can be seriously disturbed with too low a temperature, because then there is insufficient energy available for this process. Therefore you have to make sure that when watering the plants the water is approximately 23 deg C.  Oxygen is less available at higher temperatures. With cultivation on hydroponics, the nutrient solution temperature is held constant with the use of a special twin glass sleeved aquarium heater that has a built-in thermostat.