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 the 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 Emporer Aurelian made the Cult of the Invincible Sun official only in 274:
The Roman gens Aurelia was associated with the cult of Sol. After his victories in the East, the emperor Aurelian introduced an official cult of Sol Invictus, making the sun-god the premier divinity of the empire, and wearing his radiated crown himself. He founded a college of pontifices, and dedicated a temple to Sol Invictus in 274. It is possible that he created the festival called dies natalis Solis Invicti, "birthday of the undefeated Sun", which is recorded in 354 (in the Chronography of 354) as celebrated on the 25th December; (7) but no earlier reference to it exists. The cult of Sol Invictius was the leading official cult of the fourth century.
In the legions, where a policy of individual religious freedom is attested by personal inscriptions, on shrines and through votive offerings in every part of the Empire, outside the camps themselves, the only Eastern cult that was officially tolerated, probably from Aurelian's reign, and certainly under Constantine, was that of Sol Invictus. - Sol Invictus
The Cult of Mithras thus came into being after Augustine and became obsolete in or around the 4th century BC.