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Base tax
Tax value
Manpower
Trade power
Fort level
Autonomy (negative modifier)
Size of nation (negative modifier; the larger the country, the cheaper the provinces)
So as buildings are built up, basically all of the values stack to greatly increase the cost of provinces over the course of the game.
The easiest way to reduce the cost is claim, admin ideas also help a lot.
Here is the correct information:
Core cost depends on:
20 Admin Power per 1 Base Tax (not including buildings that increase tax or tax efficiency).
An overseas province of your own culture costs 10% of the normal cost to core.
War exhaustion increases coring cost by 2.5% per point.
The enemy's Hostile Core Creation Ideas, Traditions, Bonuses and Policies.
Coring costs do not depend on:
Your current overextension.
The distance from your current cores.
Your current stability.
If you are at war or not.
Costs are also reduced by:
A claim on the province reduces cost by 25%.
National Ideas, Traditions and Bonuses.
The third HRE reform, Reform the Hofgericht, gives 10% reduced cost for the emperor.
The Government Type Enlightened Despotism gives a 5% reduced cost.
Core cost reduction is capped at -90% (2 per base tax); it can never be free or instant.
As of 1.9, you can core same-continent colonies for a reduced cost of -50% in Admin Points.
Hope this helps. Good Luck and Have Fun.
Hey Kageryuu, just FYI they changed the Embassy building now. It no longer increases Hostile Core Creation. Instead it gives you +1 Diplo Relations.