As your husband learned, there is a halachic concern about immersing in hot water on Shabbat. The halachic consensus, however, is not to be particular about it. Reasons suggested for this lenient practice are because great rabbis in the past did not actively protest, because the halachic issue might apply specifically to bathhouses and not to mikvaot, or because nowadays there is a greater health concern with immersing in cooler water.
For sources, see Shiurei Shevet HaLevi 197:2:6.  
If you wish to show concern for this issue, you could limit your immersion to one or two dunks on a Shabbat mikveh night, even if you usually do more.