It's really a matter of taste. Though, to clarify, the recommended amount of time for cold brew is 8 hours - often overnight in the fridge.
I use hot water (hot as the tap will make it), and then immediately put it in the fridge. This is for a couple reasons. I prefer a little more acidity in my brew, given by a increased water temperature (It should be said that this method produces a drink that is still a lot more mild in acidity than traditional preparation). I also prefer more generous amounts of caffeine, whose liberation from the bean is dependent on both brew time and water temperature - this method producing something that is more like or traditional prep. Take it out, plunge, drink.
Cold water, used to brew french press in a cold fridge overnight tends to produce a weaker, more watery substance that , while still delicious, will not light a fire under your ass as one would expect a french press to do.