As of right now all factions are one and the same. I have yet to limit temple construction so you can have as may temples as you want in every city (1 per deity though). Might limit it to one temple per city, or one temple per tier (3 tiers).
Currently the temples purpose are a bit ambiguous, they shall both give small upgrades to the city. (Increase farm production, stronger ships, units, etc. (depending on which god) but also have a global purpose.
The global purpose is up in the air aswell, but I was thinking that you can use a priest to "worship/sacrifice" to that god, which will improve his/her happiness-o-meter to your faction, over time it goes down and if you forget about him he will punish you, but he can also bless you. Give you a monster to control or such things. (Like Pharaoh and Master of Olympus).
Haven't yet started on the gods, this demo version is just to reveal some core mechanics and hopefully spur some interest.
Funny you should mention Hegemony, it is named: Hegemony: Ancient Greece in the loading screen.