It's The Sorrows of Empire, not Sorrows of the Empire. The phrase doesn't refer to a specific empire, but to "empire" as a generic concept; it's an expression referring to the hardships, burdens, and costs faced by empires and imperialists in general. Sort of like "Uneasy lies the head that wears the crown," I suppose.
I'm not sure where the phrase originated, though. It sounds like Kipling, but I can't find any Google hits to confirm it.
It's wide-spread usage seems to from a book about American empire building from 2004 - I can't find any usage of (beyond one as part of a longer sentence) it in any book I can access via google books or any academic database I have access to.