I think some of this was present in his discussion of Constantine, but to a much lesser degree. His approach to Constantine's conversion begins in sketching the lead-up to the crisis of the 4th century by recounting the trials of the various Roman Emperors from Marcus Aurelius onwards as they struggled to keep the empire together. Concomitant to the political crises that wracked the state, there was an ongoing disintegration of religious faith—Roman citizens began to import various other faiths found on the outskirts of the empire, and everything sort of congealed together into a mass of fungible deities. Some of what I saw in the Greeks book comes through here, as Burckhardt paints this as almost a marketplace of religions where Christianity managed to win out because it was, in a sense, a better product (hard to beat eternal salvation from your earthly troubles). Even external to Christianity, though, things were coalescing around monotheism, with Elagabalus made inroads on a monotheistic cult around an eponymous god brought in from Syria and the later advent of Mithraism. Neoplatonism is also making itself felt in the 3rd century and Burckhardt shows it too is tending towards monotheism. (There was also a lot of cool stuff about Neoplatonist conjurers who would curse people by using an elaborate set of magic tricks—this too testifies to the cultural exhaustion of the time, the death of true belief and reliance on gimmick, that made it ripe for Christianity to rise). Burckhardt is definitely trying to claim here that certain pagan practices/beliefs/rites were taken up in altered forms by Christianity, but I don't know enough about the historiography of Christianity to know if there is any actual intervention happening here.

We then get a run through of Diocletian's rise to power before finally getting to Constantine himself. Unfortunately, I again don't know too much about the state of late antiquity scholarship in the nineteenth century to say whether his privileging of Diocletian over Constantine in this section constitutes something new. There is, however, a point to be made about his treatment of the question of Constantine’s actual faith. He counterposes Diocletian’s supposed unwavering, albeit pagan, faith to Constantine’s politically-motivated, almost mercenary adoption of Christianity. From the perspective of an actual antiquarian, it could be interesting whether Constantine actually believed in Christianity, from the perspective of history, however, it seems not to matter: the role he plays in history is independent of his faith (or lack of it). The section on the shenanigans surrounding the Council of Nicaea and the back-and-forth between Athanasius and Eusebius was quite funny, though.