No because no Empire has ever followed that kind of naming logic. It would also be very easy to insult other nations with that name. It implies that only we are free and the rest are enslaved or oppressed. Wether or not that is the case you do not bring it up. The Empire is named like that because the region it was formed from had that name. You didn't see Rome change its name when it expanded to other people, The British, Russians, Germans, French, and every other Empire doesn't change their name just because more people are added.
Even they didn't call themselves anything like the free peoples. It was always socialist republic, Democratic Republic, or anything else that makes them seem better. And that is only because republics and democracy actually managed to succeed in our world or else there would be no positive connotation to those words.