Well maybe he is just a great guy. I have no idea why he would want to foot the bill for the central air though. You could just chose to ignore this and hope he doesn't bring it up at a later date. I do still think you should send him a letter letting him know that in the future he really needs to contact you first before any work is done. I know that if he is just a great guy that a letter may tick him off as well but I would be worried about what he could do in the future without your consent. Did he get a top of the line one or one that was the least expensive? If you have the cash you might offer to reimburse him for the cost of a cheap one.
He pays on time and the electricity bill will go down. . . Two big positives in your favor.
Also, he fixed something w/o bothering you - saving you time and effort.
If there is a problem later on, the lease should cover your butt. You might want to remind him to keep the communication channels open and that the lease says to call you first.
The only problem would be that if he damages the property on the install or buys a very expensive AC and tries to stick it to you. Sounds like your lease has you covered. Some renters do actually treat the place like it is there home (as opposed to those that don't and trash it). I'd rather the first type.