So in a local library I found a copy of the 2007 Cambridge Economic History of the Greco-Roman World and in the environmental background of that book it describes the Roman Empire as about 2°C warmer than in contemporary times. Does more recent environmental history research corroborate this and what would it's impact have been?
This is referring to the Roman Warm Period (or climatic optimum if you're fancy). There's a science side to this question you're misunderstanding - the "world" was not 2 degrees warmer, this is just referring to Mediterranean sea surface temperature. The actual climate on land was probably more comparable to the period from 1960-1990, so pretty warm but alarmingly, you might notice, not as warm as now.
The science side is discussed quite nicely by u/CrustalTrudger over at askscience here: https://www.reddit.com/r/askscience/comments/mtf2rw/in_the_roman_warm_period_the_climate_was_2/
That's the science side, while you're waiting for an answer for the human history side this older answer from u/ladyderpette might give you some info.