I'm pretty sure you can feed it anything. Not literally anything, but for example if you fed your rabbit meats (which you might normally feed to a raptor or a behemoth or something), it'll take it. Like if you use any item on this monster food list that it'll take it. It's just that your monster will prefer certain categories, such as greens for rabbits.

Could definitely be wrong though, I'm working from memory. I did Rank 7 everything and collect every cheer so I've poured quite a bit of time into it.