Why did the arts flourish under Augustus?

Augustus came to power in Rome after a long period of incredibly destructive and bloody civil wars. He specifically wanted to use the arts to re-establish Rome as the mighty nation it once was with himself as the supreme ruler of the new empire. Thus, you can see two motives behind this new flourishing of the arts: to bring a damaged nation to a position of grandeur and to establish the power of himself as first citizen, or princeps.
Augustus famously said that he found "Rome a city of bricks, and left it a city of marble." He wanted Rome to not only resemble its former glory but to exceed it. He ordered the construction of great temples to make a point of Rome's piety. He commissioned poets such as Livy and Virgil to write epic works about Rome's mythic past. In short, Rome was now to be a shining beacon of arts and culture worthy of its past triumphs and heritage. Other wealthy Romans, eager to flaunt their wealth and stay in the good graces of Augustus, likewise commissioned their own works of art during this period.
Augustus also had the difficult task of establishing himself as the supreme leader of the nation. For over two centuries, Rome had been a republic with a strong disdain for monarchies. Augustus used visual and written arts as propaganda to legitimize his rule. Statues of him were made all over the empire, often depicting him in military triumph or as pontifex maximus, Rome's high priest.
However, a more simple answer to the question also exists. Arts flourished under the rule of Augustus because Rome was finally at peace. The period of peace and prosperity known as pax Romana, meant that more time and resources could be spent on creating artwork instead of being engaged in the long series of civil conflicts that had preceded it.