I like this idea. I also like the UV one, though both seem rather hard to balance. It'd be interesting to make it so that you could transfer the level to a UV though, only on 5* or end of line weapons (as in, heat it until 10, then roll one-three UVs with the heat, same as if you were crafting, or even make it so that you could end up not getting a UV?) Or perhaps make it so that a secondary "UV" bar is there, like nicoya suggested (As in, when a weapon reaches level 20 it has a chance of gaining one UV (only if you want it to), and then at 30 2 uvs, and then 40 3 or something. Still hard to balance.
Another thought- maybe heat is only used to level stuff for a reason? People have suggested stuff forever, and nothing has been implemented, so it might be to entice us to earn more weapons to heat?
I thought this was going to be about being able to turn heat into UV rolls. I was disappointed.
I don't really like the idea of this just because you would feel like you need to heat your stuff prior to doing new content. Also, it makes it harder for 000 to balance, since they don't know if your items are level 10 or 11.