The book refers to Augustus's government as "a monarchy disguised as a republic." What does this phrase mean? Does this disguise continue throughout the empire period?

A monarchy disguised as a republic is used to describe a society in which the people are supposed retain supreme control over the government (republic).

Augustus’s government was a monarchy disguised as a republic. Like Julius Caesar, Augustus ruled by centralizing political and military power. Yet he proceeded more cautiously than had his patron: Augustus preserved traditional republican offices and forms of government and included members of the Roman elite in his government. At the same time, though, he fundamentally altered the nature of that government. He accumulated vast powers for himself and ultimately took responsibility for all important governmental functions. He reorganized the military system, creating a new standing army with commanders who owed allegiance directly to the emperor—a reform that eliminated problems caused during the late republic by generals with personal armies. He also was careful to place individuals loyal to him in all important positions.

After Augustus, the Roman Empire continued to grow to the point that it surrounded the Mediterranean. For example, in Byzantine Empire that was founded after the Rome Empire declining, he emperor decided to divide the empire in half because he thought that it would make it easier to govern.  Later on, while civil war ravaged the western half of the Empire, the eastern half of the empire was pretty stable and so Emperor Constantine decided to create a new capital at the former Greek city of Byzantium, which he renamed Constantinople (present-day Istanbul).  

References

Chacha. (n.d.). Retrieved from http://www.chacha.com/question/the-book-refers-to-augustus's-government-as-%22a-monarchy-disguised-as-a-republic.%22-what-does-this-phrase-mean%3F-does-this-disguise-continue-throughout-the-empire

Jerry Bentley, H. Z. (2009). Traditions and Encounters . McGraw Hill.