Do US universities allow a professor to work in a brothel?
It's typical for US universities to require disclosure of certain types of outside activities due to concerns with conflicts of interest and to comply with federal guidelines:
https://www.unlv.edu/research/coi#guidelines
There are also typically clauses that don't allow outside activities to negatively impact work with the university.
I think it's likely that this would fall into a grey area where there is clearly no direct conflict of interest for the purposes for which such policies typically exist, but a broad interpretation could allow a professor to be removed if those outside activities were considered too much of a distraction. If students were aware of the other position or could be clients, that might be seen as more of a problem.
That might lead to a legal challenge but I'm not aware of any related legal cases related to brothel employment by professors, and broadly US employers have a lot of leeway in deciding who they employ. There is this story of a journalist-stripper-professor who was fired from her journalist position when her employer learned about her blog about stripping, but apparently not her lecturing position.
Note that having outside interactions with students doesn't need anything as potentially controversial as sex work. There are lots of ways students could potentially have a conflict of interest with a TA or professor if they have some financial relationship: if they hire them as a tutor, if they work at a bar/restaurant and are left a tip, if the students' parents buy the professor's house, all of these could raise a potential conflict of interest situation.