To start off, it's difficult to determine when exactly Rome, the city, was lost. Is it in 476, which is the commonly accepted date for the fall of the western Roman Empire? An argument can be made for this not being the case, as the Kings of Italy still considered themselves to be de jure vassals of the Roman Empire, which by that point was represented by only the eastern half. So Constantinople still exerted a certain amount of control over Rome, certainly until the invasion of the Ostrogothic kingdom by the Roman Emperor Justinian in 535.

However, for the sake of the question let's presume the end of the western Roman Empire in 476 to be also the fall of Rome. The importance of Rome as the socalled capital of the empire had been greatly diminished by 476, while it undoubtedly still held some cultural signifance, no emperor had really considered Rome to be the seat of their power for the hundreds of years prior to the collapse of the western Empire. As a matter of fact, Ravena, located in the more strategically important and easier to defend northern Italian swamps, was considered to be the de facto capital of the western Roman Empire during its collapse in 476.

The signifance of Rome as an important city of the Empire began to decline in 286, when (western) Emperor Diocletian moved his capital from Rome to present day Milan. It was further amplified by (overall) Emperor Constantine in 330, who designated by then Byzantium to be the new center of the (Eastern) Roman Empire due to its great strategical and economical significance. Not only did Byzantium guard the access from the Bosphorus strait into the Black Sea, which was considered to be a very important sea because it controlled a lot of the trade coming into Asia, but it was also located in a strategic location. From Byzantium the Roman legions could quite easily reach both the Persian Frontier and the northern frontier along the Rhine, whereas Rome was located far away from any military front. Constantine thus named the city of Byzantium after himself and it was henceforth known as Constantinople. As by this point in time the Eastern Empire was without a doubt the stronger of the pair, a lot of importance was ascribed to this move of the capital to Constantinople.

The fall of Rome was not considered to be a huge loss, certainly it became a goal of the eastern Empire to reconqueror Rome because of its cultural signifance, but it had not been their capital for almost 200 years, they ascribed much more importance to Constantinople as the cultural, religious and military center of their empire.

I've tried to answer the question to the best of my abilities, but I have to admit I wasn't 100% sure what was meant by ''Rome (the empire)'', as the western Roman Empire barely had time to mourn the loss of their territories before they disintegrated entirely. I've therefore spend more time on the eastern empire than the western empire while answering your question.