I understand that the notion of state in the specified period was different from the modern one and maybe was gradually changing over time. However, I still believe that contemporary sources occasionally had some need to refer to the political organization nowadays called "the Roman Empire."
I presume that the Early Roman Empire (principate of 1st-3rd centuries AD) stubbornly continued referring to itself as the Republic of the Roman Senate and of Roman people, the idea that I assume, often was meant behind the abbreviation of S(enatus)P(opulus)Q(ve)R(omanus).
If so, was it the only way to refer to the state? Did it change with the transition to a seemingly more authoritarian political regime in Late Antiquity (from Diocletian onwards)? Did the spread of Christianity and increasing acceptance of it by the ruling elite influence the way contemporary sources mentioned the state?