Even if you rename a Studio, it does not change the URL of the Studio. This allows any invite links you have already sent to remain valid since they use the URL that corresponds with the Studio's original name.
However, you can create a new Studio with a different name, which will be used as its URL. For example, if you call your new Studio: Possible Barefoot Moose?, its invite URL will be something like: https://riverside.fm/studio/possible-barefoot-moose?t=0123456789abcdefghij.