Still working on this one, but it looks as though the game brings together your scores from all your save files. I completed the game on normal, started a totally new file on easy, and accessing the phone in the Sheep lists all my awards on either save.
You can also play all levels that you’ve previously completed through the phone, which saves dealing with cutscenes and the like.
You need to get gold awards for stages on easy, normal and hard, on both classic and remix.