Hm. Not directly, but let's give it a try. Roman heavy industry could pollute on an impressive regional scale, perhaps including traces of lead that appear all the way up in glacial ice in the Alps. But it's more difficult to distinguish the carbon impacts of individual empires—or, perhaps more accurately, of the imperial subjects living within those empires. Recent studies have typically focused on how the environment shaped the Roman Empire, rather than vice versa. That's in part because there isn't clear evidence that ancient empires or their imperial directly contributed to environmental change through carbon production.
That said, premodern people did substantially alter the global ecology, primarily through deforestation for grazing and farming. Plants, perhaps surprisingly, make up the bulk of all global life. Scientists estimate that there's about 550 gigatons of carbon (GtC, sometimes also written PgC for petragrams of carbon) currently distributed among all the living creatures of earth, and they think that about 450 GtC of that belongs to plant life, or about 82%. But a survey of global land use shows that these numbers fall far short of the global potential. Scientists estimate that the earth could actually support about twice as much plant life, or about 900 GtC, if it weren't for human intervention. And they say that at least 20% and likely closer to 80% of this loss occurred during the preindustrial period, as vast regions of the world were cleared for farming and grazing. To reframe these numbers somewhat, that means that before the year 1800, preindustrial humans wiped out up to 425 GtC of plant life, or perhaps 40% of all global life, reshaping the global environment to allow the growth of human civilization and agriculture instead.
It's hard to pinpoint when these changes occurred, and they probably reflect a pair of processes. On the one hand, humans have certainly cleared vast amounts of forests. For example, Europe no longer has enough oak trees to build a cathedral roof, which is why that's not an option for replacing the roof of Notre Dame which burned in 2019. On the other hand, since humans adopted agriculture right after the end of the last Ice Age, or about 10,000 BCE, it's likely that this is not just about deforesting established woodlands but also about preventing preventing woodland growth to begin with.
The cumulative effect of this was substantial, but it's hard to parse the effects of individual societies. The Roman empire seems to have benefitted from slightly warmer temperatures, which would have helped facilitate intensified agriculture and population growth, but there seems to have been a lot of global climatic variation that obscures things. Good times for Roman agriculture were matched by bad times elsewhere, and vice versa, so it's hard to isolate the effects of the Roman Empire from global data against this changing background. Furthermore, the Roman Empire overlapped with the Han dynasty in China, which makes it difficult to discern which empire contributed what to global data sets.
We see a much clearer case in the 1500s, when diseases like smallpox and the pressures of forced labor wiped caused a sudden and dramatic depopulation in the Americas. This serves as at least one rough benchmark for making assumptions about the deeper past. Researchers estimate that the population of the Americas was around 40 to 70 million people in 1500, but that it dropped to 2 to 7 million people by the year 1600. That’s a decline of between 90 to 95 percent. Farmland across the Americas was abandoned and reverted to woodland. These new plants captured tons and tons of carbon out of the global atmosphere. Ancient air trapped in glacial ice reveals that the global environment had CO2 levels of about 282 ppm in 1500, and that it dropped to 275 ppm by 1600. This doesn't seem like much by today's standards (we're at 417 ppm today, which is up 30 ppm since just ten years ago), but it was a pretty big change by pre-industrial standards. There's no easy way to show this on Reddit, but if you scroll down to the second chart on this page (titled Global CO2 Levels), you can click and drag from about 1400 to 1800 to see this change in smaller temporal scale.
Can we bring this back to the Roman Empire? Kyle Harper, one of the most provocative (and controversial) thinkers on the Roman environment, has proposed that the population of the late Roman Empire was about 70 million people. That's similar to the high estimates for the Native American population in 1500. Roman land use probably paralleled in very rough terms the land use of Native American empires like the Aztecs and Incas around 1500. If their collapse reduced global CO2 levels by about 5-10 ppm, then it seems reasonable to assume that the Roman Empire had an inverse effect of similar scale, perhaps raising global CO2 levels by about 5-10 ppm globally. (Again, that's on par with the global change we're seeing every 2-5 years today—yikes!)
Roman industry and urbanization probably contributed to this somewhat, but the more significant factor was almost certainly the challenge of having 70 million mouths to feed. The Roman Empire might have increased this burden somewhat by encouraging population growth through intensified agriculture, but this was likely to occur anyways thanks to warmer local climatic conditions. Further, the Roman population mostly relied on areas that had long been settled and had already suffered extensive deforestation due to farming and grazing, so we probably shouldn't lay too much blame on the empire itself.
TLDR We have no real means to estimate the carbon footprint of individual Roman subjects, but they probably didn't contribute much carbon directly to the environment. Instead, the Roman population survived thanks to massive deforestation efforts to clear pastures and farmland to feed 70 million people. Much of this deforestation likely predated the Roman Empire, and it necessarily contributed to elevated global levels of CO2. Although it's impossible to calculate the precise scale of these effects, they would have been almost certainly negligible compared to current annual changes.
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I should perhaps note that one earlier study (pdf) proposed a 13% reduction in global carbon emissions during the decline of the Roman Empire, largely due to depopulation and reforestation during pandemics. Researchers today are skeptical (New York Times; possible paywall) of the depopulation models behind these numbers, since they make false assumptions that the Black Death should be our template for understanding all premodern pandemics.