Already Googled: various combinations of Classical Rome+baths+slaves
A couple of sites I came across said that "slaves weren't allowed to use the baths" but didn't elaborate or give any references.
Now, I know that slave-ownership went way down the social scale, so that thousands of free Romans owned household slaves who didn't have posh enough houses to have their own baths - they went to the public baths. I can perfectly understand that they wouldn't have wanted to have slaves enjoying the hot plunge pool with them, but Romans took as axiomatic that the steam/oil/scrape/plunge process was necessary to cleanliness, so much so that, for example, every private soldier in the Roman Army had bath-house facilities - it was clearly unthinkable that squaddies should be expected to make do with hot water and a flannel. So I can't imagine that they would have been happy to have cooks, nannies, maids etc. in their households who were not able to get clean.
Public baths were open to men and women at different hours, or different days. Were there possibly times reserved for slaves? Or were there separate cheap establishments for slaves, where they got steamed and sluiced without the benefit of posh decor?