Is there anything halachically wrong with entering a hot tub on Shabbat inherently? Not for hygiene just leisure purposes, and not adjusting the temperature, just entering it?
Our Sages prohibited bathing one’s entire body – or the majority of it – in hot water1 [on Shabbos], whether the hot water was in a container or [collected] in the ground,2 even if [the water] was warmed on Friday.3 [This prohibition was instituted as a safeguard] because of bathhouse attendants who would heat [water] on Shabbos and say that they heated it on Friday.4 True, Jews are not suspected of willfully desecrating the Shabbos.5Nevertheless, [the attendants] would [build a fire by] placing wood under [vats of] cold water while it was still day [on Friday], just before nightfall, and [the wood] would burn the entire Shabbos, heating up the water above it. [Doing so] is forbidden because [of our Sages’] decree: lest one forget and stir the coals after nightfall.6 Therefore, [our Sages] prohibited any person from bathing with hot water even outside a bathhouse cared for by an attendant. [It is] even [forbidden] to pour [hot] water over one’s body to rinse oneself,1 even though this is not the manner in which one bathes in a bathhouse, for in a [bathhouse] one submerges his body and limbs into the water, [and in this instance, one is pouring the water over his body
Based on the reason provided the distinction of being for leisure vs hygiene would not matter. On the other hand for tevilla in a mikvah going into hot water would be permitted. See this link for a discussion of being for a mitzvah... https://outorah.org/p/60560/