Can I send my mentor's rec to another school?

My research mentor wrote me a college rec and sent it to my first set of schools. He also sent me a copy (as a word document) and said it was “for your records.” Can I just send this letter to another school or is that weird? I just don’t want to bother him (he’s filled out a lot of forms for me). I would have to change the name of the school in the rec.
Thanks! I’m just not sure what’s appropriate.

He should send it out, with the name change. It is never appropriate to modify a letter of rec even for something that seems innocuous like the name of the school.