It sounds to me like Airbnb really doesn’t have anything to do with your rental, although if you inform them that the listing is inaccurate they might ask the host to correct it.
Another issue may be “what does the term Air Conditioning mean?” In a legal sense it may not mean what you think it means.
What is critical here is what your separate rental contract says. A court will seldom consider anything outside the “four corners of the contract,” so unless it specifies a particular type of cooling system you have no recourse.
If the house is unbearably hot the first step would be to politely ask the host to do something about it, either repair a malfunctioning existing system or possibly install a window unit in the bedroom due to the current unusual weather.
Or you might consider buying a cheap unit yourself and offer to leave it there if the host reimburses you for it.
Until recently our Ozark mountain cabin really didn’t need air conditioning if you opened everything up after sundown and closed up again before noon. Now it would be quite uncomfortable without a refrigeration system and so we installed one, which unfortunately is abused by people who don’t have to pay the electric bills.
