That doesn’t make sense except that it would absolutely increase the blood flow to the area and probably draw the heat out faster.
Interesting.
I suspect that restaurant kitchen burns frequently involve hot grease or hot, greasy foods.
In such cases, it's probably important to flush / irrigate the skin in an effort to sweep away / rinse off foreign matter / contaminants.
Hot (not scalding!) water would probably be more effective to do this than cold water.
Cold water might cause the grease and oils to actually congeal in situ - thus worsening the healing.
So the hot water wouldn't help to cool the burn site, but it would at least be more effective in eliminating contamination of the wound.
Probably best to first clean with hot water, then switch to cold water.
Just my 2 cents!
Regards,