Would an Unfettered Eidolon work as different creature types?

[Reksew_Trebla]

The Unfettered Eidolon has specific rules for how to advance it. It has an amount of HD at each CR equal to the amount in this chart. Would this still be true if you changed its creature type to, say, Fey, and used the HD per CR for Fey on that chart instead of Outsider?

I'm not going to change it to any creature type that doesn't make sense though, like Animal, but I think this could be interesting to use to make new creatures.

What do you all think?