Thermae Romae is an anime where a bathhouse architect from the Hadrian period of ancient Rome finds inspiration by time-traveling to modern japan and copying their bath-house culture. The protagonist starts with simple things like paintings of mountains to adorn his ancient Roman bath-houses, but soon moved on to more extravagant things like fruit-flavored milks, heated floors, slave-powered massage chairs, and water slides.
I understand that Hadrian did in fact have a private bath built to mimic Egyptian waterways, crocodiles and all, in honor of his deceased lover.
So my question is, how elaborate were bathhouses, either public or private during Hadrian's reign?
Edit to add the show mentions that slaves had, by law, the same access to ba to houses as full citizens, is that true? Also there's no mention of women's bathhouses, except a brief mention that different times were reserved for women and men, would women have access to the same amenities as men?
Second edit: at the end of the show >!The Senate conspires to destroy the bathhouse architect protagonist because the popularity of his bathhouses are seen as supporting Aurelius Caesar.!< Is there any truth to this? Or the idea that a popular entertainer would become such a target?
Ok, I can try to give an overview. We actually do know a lot about Roman Bathhouses because quite a few of them have survived quite well, public ones as well as private ones. For big public ones, there are well preserved examples in Bath (Great Britain), Lepcis Magna (Libya), Carthage (Tunisia) all of them completed in the Second Century AD, shortly after Hadrians Reign. For Rome itself, the best preserved public baths are those of Caracalla completed in 235 AD. A short, but useful summary, a ground plan of Caracallas Thermae and a few pictures can be found here:
https://www.worldhistory.org/Roman_Baths/
In Pompeji, the city destroyed by the eruption of the volcano Vesuv, there are two public baths that survived destruction. In addition to this, there are more Roman Baths next to military camp sites (for example in Weißenburg, Germany) or in larger private houses or guest houses (reconstruction of on active today in the Archaeological Park at Xanten, Germany).
Floor heating was a standard feature but used in the rooms with the hot water basins. The baths held nearly always several rooms with water basins, and they used to switch between cold water, moderately warm water and hot water. In large baths there even could be some kind of sudatorium (sweating room). On the link I gave above, there is a ground plan for Caracallas Thermae. You can easiliy see the design is mirrored. For these, it is believed that one side was reserved for women, and the other for men.
But these thermae are exceptionally big. In smaller towns with smaller installations there would be one set of rooms. And yes, there would be different times of bathing for women and men. We do have a written source for this, a town law of a small town in Spain, were this custom and the prices were written down. In this text there is mentioned that women had to pay more and bathed in the morning - which meant they got the cleaner water...
Private baths are a lot smaller, because ther was no need to fit in hundreds of people at the same time - but the features of several rooms, partly with heated floor and walls were the same.
For further information, you might want to look into: Garret G. Fagan, Bathing in public in the Roman world, 2002.
Also there is a bibliography of literature on different Roman baths by Hubertus Manderscheid: Ancient baths and bathing. A bibliography for the years 1988-2001, Journal of Roman archaeology supplementary series 55, Portsmouth 2004