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So, in the new canon, before the Galactic Republic was "the Old Republic."

If you're writing a story during that time period, I assume it wouldn't be called "the Old Republic," right?

So, would it just be called "the Republic," or was there probably a name in front that we're not aware of?

Yeah.

Who would have named his government with "Old" in the name at the time of creation? I guess when it was intact, it was called just the Republic in those times.

The RotE Republic was also referred to as "the Old Republic" after the formation of the New Republic.