I have 2012 Range Rover Supercharged since few months and I also noticed this similar loud fan noise almost all the time now. I assume that in this peak summer when AC condenser is working overtime so the engine fan or AC fan must be working hard to keep the engine cooled. Have you experience this in a cooler month, as in this hot months, when you start the car inside of the car is, even more, hotter than outside and AC cooling must be working hard in summer months, to bring the cabin temp down asap.
Secondly, when you cold start any car, revs, and fan is always high (1500 rpm) and then settle down in 15-30 seconds in all Land Rovers to 800-900 RPM.
However, you can try this that start the car in the night and drive for 5-10 minutes with AC on. Park it outside in open area and step outside to listen if AC fan speed is slowing down after a certain interval or not. It should cut off for a few seconds in between and then restart again like 30 sec ON - 5/10 sec OFF.
@Barry and @shadow79 can advise further on exact intervals.
TBH, I wouldn't worry much if the dealer has inspected it and said it's fine.