The best replacement for hegemony is empire.

The best alternatives to hegemony are rated as following:
★★★★★  empire
★★★★★  survival
★★★  ideology
★★  dominance
★  imperialism
★  oligarchy
  globalization
  unipolar

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Speaking about hegemony, one probably shouldn't forget about hegemon.

Other search terms related to hegemony and their ratings are:
★★★★★ hegemon
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