Question: Can one take a cold shower on Shabbos?
Answer: The Gemara (Shabbos 39b) writes that chazal decreed that one mustn’t wash one’s entire body with hot water on Shabbos, as they suspected that if it would be allowed, people would heat up water on Shabbos. The Shulchan Aruch (OC 326:1) writes, however, that one is allowed to wash one’s face, hands and feet with water that was heated before Shabbos.
The Shulchan Aruch (OC 326:4) does allow one to pour cold water over one’s whole body. Nonetheless, the poskim write that the ashkenazi custom is to avoid showering on Shabbos. The Magen Avraham (OC 326:8) writes that one can’t dip in a river as we are concerned that one may carry (outside of an eruv) or squeeze water out of one’s hair, etc. (See Shut Maharil 139; Mishna Berura 326:21).
R’ Moshe Feinstein (Igros Moshe OC 4:74: rechitza 3; 4:75) writes that while the reasons for this minhag may not apply to taking a cold shower in one’s house, one shouldn’t do so under normal circumstances. Only if one is particularly uncomfortable (mitztaer) such as during a heat wave, may they do so. R' Moshe Stern (Be'er Moshe 6:73) writes that one doing so under such circumstances should cover up their hot tap and avoid wetting their hair.R’ Akiva Eiger (OC 326:1; 5) even allows one to use hot water provided it was heated before Shabbos (e.g. from a Shabbos kettle), though one doing so must ensure that they don’t pour the water directly from the kettle into the cold water.