This chart is neat and totally relevant!
It lists several materials' compatibility with hydrogen peroxide. Even at 10% solution (just about anything you'd buy at a store is going to be 3%), silicone and hydrogen peroxide are ranked as being "Excellent." That is, it has no chemical effect!
Natural rubber (which is what the original Keeper is made of) is listed as having "good effect," which could represent "slight corrosion or discoloration."
So there you have it!
**ETA Personally, I soak my Mooncup UK cup in a 1:1 water to hydrogen peroxide solution and it leaves it all sparkly, clean, and fresh.