Any reason for not turning it on in the air (with the CAT 3 exception)?
There are possible advantages - so why not?
Errrr ... wastes fuel, wears out, might catch fire... I am not sure what the "advantages" are?
On a CAT 3, or other approach, I presume it is running in case of a generator failure? If so, I would be surprised if the transfer to APU would be "seamless", and might be enough to trip off enough systems to cause you to GA anyway?
I flew one type, albeit not airliner, where the APU had successfully been used for years on the ground. The moment is started being used airborne, or rather during landings, it experienced a high failure rate