Very good ideas from people on this! Here is another one.
The gas and/or electric company may be able to provide data for the dates in question, and other dates for comparison. The host may be able to request this and receive written documentation. With the smart meters now it should be easy to tell. The property manager may have been mistaken or it may have come on mysteriously after you left. Consumption data should clarify the situation very rapidly. I would ask the host to provide documentation of usage.
I can see hourly electric usage on my SCE account up to and including yesterday. The gas company website only shows monthly readings.
Unless the rules have changed, the host must report a problem within 24 hours. This means he may have reported it based only on the manager info and report of "bath water" temperatures. Airbnb will know if he has reported problems like this before, in which case he does need to document, and provide a solution. If this is a pattern with him, then that might indicate a different story. Was there anything about this in his prior reviews?
Best of luck with this predicament, and thank you for posting. It is good for us as hosts, but maybe more so as guests. Sometimes we might have to be a little careful to protect ourselves, especially where there is a fixed extra charge on offer.
I would like to add that last year I had guests in my own home for a whole house rental. They caught an antique table on fire, and then did not want to pay for the damages which were about double their $700 deposit. We got a nasty letter from a lawyer who was related to the group that stayed. I called Airbnb and they had received the same letter the day before and were all over it. They handled the whole thing very gently and professionally.
The Airbnb rep said the host guarantee would cover what the security deposit did not, but that they would try to recover it from the guests first, and that they would handle the lawyer. I am not sure who actually paid, but it was all done in under a week. Airbnb did go to bat for us. I did not have to do anything apart from submit photos of the damage, of the repair estimate, and some comparisons to show replacement value. It was bonafide, so easy to document. I expect your host will also have to document somehow.
I am mentioning this because I think Airbnb will take good care of you on this. Most people are honest, so I expect it is a misunderstanding of some type. It is odd that he offered to heat it, but you declined, and then he said it had been heated. Plus I doubt the manager was there 24/7. Very hot weather will heat a pool too! Let us know what happens, but I expect it will work out smoothly.
Thanks again for posting.
--Lisa