In a 1943 article in the Journal of Roman Studies, MP Charlesworth notes that for the Romans pius, felix, and invictus were intimately, even causally, connected: “because the Emperor is pius the gods will render him felix (for felicitas is their gift to their favourites) and his felicitas is best demonstrated in his being invictus.” The gods are especially pleased with an emperor who collects more worshipers: Conquest is a demonstration of piety not only because victory is a sign of divine favor but because a conqueror is a forceful evangelist for his gods.
This structure of concepts changed little when Constantine converted: “Felix may be dropped, but Felicitas vestra can remain as a style of address to the ruler who restores and ensures a felix saeculum. Invictus perhaps smacked too much of Mithras and of pagan deities for the new fashion, and so is replaced by victor or victoriosus or victor semper. Pius the emperor (in theory) always was, and continued. The same words are used, and even though there is development and new accession of meaning and significance, the root-idea changes little. Augustus had restored Roman religion and brought back the pax deorum, because he recognised that on the benevolence of the gods the life of Rome depended. Constantine favoured a new religion because he was convinced that it worked, and that the power he had invoked would bring victory and success not only to himself (as he had proved) but to the whole Roman State: two things therefore were essential (in his mind) for this ; one, that the Christian clergy should be free to carry on the prayers and worship whereby they secured the favour of Heaven, and the other, that the Christian people must be united and free from discord in their worship (for divided or partial allegiance must be displeasing to the divine power). Both these ideas are typically Roman.”
Christian Constantine remained a Roman Christian emperor.