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Modern jets really don't work like that though. There's not one engine that controls the air conditioning, and one that does not. The engines provide bleed air to the pneumatic system, which in turn provides air to the air conditioning packs. As long as the pneumatic system is provided enough pressure to run the packs, either engine can indirectly support the air conditioning.
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Depends on the jet I think. I know that the 737-300/700/500s run the a/c through the bleeds off the number two. I've been told this by several SWA captains, and the reason they start the number two first is "for passenger comfort and cooling." As far as taxi-ing in with the number one in Orlando, the APU can also run the packs. Our guys sometimes taxi in with only the number one since the cargo doors are on the number two side. This allows us to get in and start off-loading about 20-30 seconds faster. I think there might be a slight edge in fuel saving as well.