Well, it wouldn't be known as Rum (or Rûm) for the same reason as Turkey isn't known as "Türkiye". It would have to be anglicised as "Rome"/converted in various languages into the exact translations of Rome and depending on your views on how the new state would be received, that could create problems with lack of acknowledgment by the rest of Europe (and Italy in particular).
People could call it "Rum" to distinguish it from "Rome."