There was some opposition, but not much in the grand scheme of things. There are several reasons for this:

1. Octavian established controls over all the traditional avenues of power after Actium. He made sure that he was awarded a constant and renewable imperium over Spain, Gaul, North Africa, and all the other places where legions were stationed. He had learned from the previous wars that any place that could potential have armies or conflicts should be in his "permanent" control. This prevented anyone else from concentrating enough military power (under the premise of serving Rome) to pose a threat. In the early years, he also held one of the two consulships often, sometimes year after year. He essentially eliminated the route by which ambitious young Romans (like himself or Caesar) might go up the ranks and into power.

2. He carefully cultivated favor with the urban plebs of the City, who were politicised and volatile. Octavian's securing of Egypt was important for this goal, as that country's grain was organized into a nearly-constant stream of food ships to the capital. He also (as he tells us himself) spent a veritable fortune on improvements and beautification of the city. And threw lavish games and festivals, of course. All this to keep the plebs "on his side."

3. He established and maintained a monopoly on prestige display. No other Roman was ever allowed to put on a display of military virtue which could rival Augustus' own. There was simply no route and no permission to publicly announce one's accomplishments in the Roman venue any longer.

4. He was awarded (or he arranged for himself to be awarded) imperium maius in all places--that is, "greater" imperium. This meant that in any Roman place, no matter who was governing, Augustus automatically overrode that person in rank if needed. In practice, he almost never utilized this "power", but the implicit assumption was: if you get up to trouble in your province and start to amass power (as Caesar had done in Gaul), Augustus had the "legal" mandate to intervene.

5. He kept the soldiers happy and loyal to himself, permanently. After Actium, this meant keeping his promise to get them settled in new colonies with their pensions and good parcels of land on which to retire. And there were tens of thousands of them. He did this (thanks in some part to Egypt's coffers), creating dozens of new Roman colonies in the process in places like Iberia, Gaul, and Numidia. He also reorganized the way soldiers were payed, both during service and upon retirement. A fund was set up for this purpose (like a modern social security system). This tended to shift the soldiers' loyalties away from individual generals and back to the state as a whole, where the Emperor was permanently ensconced (more or less).

6. The world was tired. It had been nearly a century of more or less constant conflict. The turmoil of the Gracchi brothers, then the Social War, then the bitter conflict of Marius and Sulla, then Sulla's bloody dictatorship, then the Civil War of 49 between Caesar and Pompey, then the Civil War between Octavian and Antony. Roman society had been turned on its head, countless thousands had died, and by 29 BCE there were few people left alive who remembered a time of relative peace and stability. Octavian was simply the last man standing in a long series of struggles to "fix" the Republic, and he was smart enough to learn the lessons of his predecessors and take steps to end the cycle of violence. Most Romans, the elite included, were in the end seemingly happy to have it all over and accept Octavian as a savior, and the new "princeps" for his part was no Sulla, and at least put on a face of dealing fairly and wisely.