Goverment types

Hegemony? Imperium? Or are you looking for what kinds of bonuses and penalties such an empire would get, not for names for it?

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Historically, Empires can have Senates. Look at the Roman Empire. While it had an Emperor, it also had a Senate. The Senate's power was greatly diminished from when it was a Republic (yes, we're still talking about the Romans), but it still had some pull with the people, as well as many domestic duties.

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How about Feudalism or Confederacy?

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In the case of the Roman Empire, the Emperor was emperor of all territories except Rome. In Rome, he was simply the first citizen and the Senate ruled Rome. However, in his position as first citizen, the Emperor was also a senator (and the most senior at that) and his influence was often so great that nobody dared challenge him within the Senate. Also the Legates (Commanders of the legions) had to be senators as well, so a senator was a public officil who in certain cases was also a very senior military officer.

This is different in SW where a senator is simply a public official, at least ex-ufficio. The senate itself voted to create the institution of the emperor (and transformed the nature of the republic into the empire). A few years later, the emperor dissolved the senate by means of an imperial decree. Although the emperor left the details (day to day running of the empire) to his governers and military officers, in theory he his words were law in the form of imperial decrees. All public and military offices ultimately existed at the pleasure of the emperor and he could both bestow and reclaim authority to his subjects.

In SW, therefore, we have the empire where initially there was a legislative body (the senate) which could be over-ruled by the executive (the emperor). A few years later, the legislative body was simply dissolved and the empire was run solely by the executive (the emperor and his chosen assistants). In both cases, you can call it an "empire".

The difference between empire and an imperium is non-existent, at least today. In Roman times, Imperium meant power, so a general had an Imperium (and could be called imperator) over his armies. Likewise, a magistrate had Imperium (and he was also an imperator) over his court. The Emperor had power (Imperium) over all territories except Rome so he was Imperator of all territories except Rome, which was the Imperium of the Senate.