They’re cosmopolitan like Alexander the Great, whose Hellenistic empire was the original “cosmopolis” while also of course being super imperialist. Or like the Turko-Persian empires of Central Asia.
None of those were egalitarian. But nor did they (or could they) foster a strong sense of “homeland” identity as a means of holding things together.
The Hegemony’s ruling class (the Theurges) as well as its administrative class (priests and officials) come from all Archonties. Yes, Karagon is over-represented and versions of its language and culture are valorized as the basis for cosmopolitan identity. But that’s empire for you.
Edit: see here for my earlier account of what cosmopolitan means in this context – Napoleon rather than coffeehouses.