Yes Phoebus Apollo was the sun god during the time of Augustine at Rome!
The worship of Apollo was widespread not only in Greece but also throughout the ancient world. Shrines could be found in places from Egypt to Anatolia (now northwestern Turkey). The Romans built their first temple to Apollo (Phoebus) in 432 B . C ., and he became a favorite Roman god. The Roman emperor Augustus was a devoted worshiper because the battle of Actium, in which he gained political supremacy, was fought near a temple of Apollo. - Myths Encyclopedia.
But how do we reconcile this with the Roman Cult of Mithras or the Sol Invictus of Rome?
The Roman deity Mithras appears in the historical record in the late 1st century A.D., and disappears from it in the late 4th century A.D. Unlike the major mythological figures of Graeco-Roman religion, such as Jupiter and Hercules, no ancient source preserves the mythology of the god. All of our information is therefore derived from depictions on monuments, and the limited mentions of the cult in literary sources.
The temples of Mithras were always an underground cave, featuring a relief of Mithras killing the bull. This "tauroctony", as it is known today, appears in the same format everywhere, but with minor variations. Other standard themes appear in the iconography. - The Cult of Mithras.
The Roman Emperor Aurelian made the Cult of the Invincible Sun official in the year 274 AD:
The Roman gens Aurelia was associated with the cult of Sol. After his victories in the East, the Emperor Aurelian thoroughly reformed the Roman cult of Sol, elevating the sun-god to one of the premier divinities of the Empire. Where previously priests of Sol had been simply sacerdotes and tended to belong to lower ranks of Roman society, they were now pontifices and members of the new college of pontifices instituted by Aurelian. Every pontifex of Sol was a member of the senatorial elite, indicating that the priesthood of Sol was now highly prestigious. Almost all these senators held other priesthoods as well, however, and some of these other priesthoods take precedence in the inscriptions in which they are listed, suggesting that they were considered more prestigious than the priesthood of Sol. Aurelian also built a new temple for Sol, bringing the total number of temples for the god in Rome to (at least) four. He also instituted games in honor of the sun god, held every four years from AD 274 onwards. - Sol Invictus (Wikipedia).
The Cult of Mithras thus came into being after Augustine and became obsolete in or around the 4th century AD.
(Crossposted on Mythology.SE)