If its totally empty, then off. Other than that it doesn't matter.
EDIT: :think: Wait aminute, the new ones have only one cap... that means teh whole system should be under pressure with a hot engine. So, if the engines hot, wait till it cools off to add. Once the cap is open though, it shouldn't matter if its running or not.
if its off though how high do you know how to fill it, since the fluids can expand when the engine is hot and go past the "full" mark if you fill it up to that level when cold? BTW< I am referring to the coolant resevoir tank
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