Yechiel Michel Epstein,
Translation:Arukh ha-Shulchan…
“ Section 3 Though as a matter of law there is no prohibition against washing with hot water on these other three fast days, nevertheless the custom is to consider it prohibited, with the exception of Friday, when one can wash in honor of the Shabbos (Bach) . This is, however, no custom to prohibit washing with cold water, and one may do so. One may also sit in steam room [bathhouse] when it is not pleasure that is intended but rather the removal of the discomfort of sweating. ”
