I would insist on seeing the invoice for the new AC and writing the tenant a check. From what I understand a new AC is considered an improvement to the property and you can begin getting depreciation value the year in which it was purchased when you file your taxes. You won't be able to do that if your tenant is the purchaser. If you had just fixed the old one it would have been a repair and all of the repair cost could have been deducted. A new AC is great but your tenant should not have purchased it with or without your knowledge.
I know he shouldn't have purchased it. I'm well aware. He did give me the invoice, the work orders, the post-inspection report and everything. I don't really have the money to just pay him outright for this.
I wouldn't worry about it, The T has been good paying and not a problem. He was made aware he shouldn't have done it. I'd make a note a a Memo for record in the file on the house if it should become a problem when he leaves or at some point. I think we all understand the LL would have wanted to make the decision of brand and all of that, but that's water over the bridge. I wouldn't make a huge issue of it at this point.