I don't think I have ever seen this during the 100 or so high school games I've watched. I might have seen it during the first year of kid pitch several years back. On the surface, it sounds like an absolutely horrible idea. Unless the kid is throwing BP type stuff, having him leave the mound only to return later (20-30 minutes later after thinking he was done pitching) has all sorts of pitfalls. It sounds like your kid did not have a very stressful outing #1 (but was pitching with runners on base - but may have been enjoying a huge lead already), but if the coach does the same thing in a tight game with lots of high stress pitches, then I would be careful.
Not a fan of icing, but if this keeps happening, I'd suggest your kid tell the coach he needs to ice after every outing to keep from getting called back out to the mound. Either that or refuse outright.