"To the angel of the church in Pergamum write, 'These are the words of him who has the sharp, double-edged sword. I know where you live where Satan has his throne.'" (Revelation 2.12)
Early Roman emperors like the first emperor Augustus refused to be deified or to be called gods. Later, it was felt to be politically useful, and inevitably they soon started believing they were actually gods.
To reinforce this idea the people were required to burn a pinch of incense to the bust of the emperor and say, "Caesar is Lord" as a test of loyalty. This became a unifying principle in the Roman Empire. But it was also a simple and easy way to discover who were the Christians in the towns and villages for whom only Jesus was Lord because they would refuse to do this.
The Emperor's bust set up in the town square therefore became the place of condemnation for Christians, where "Satan has his throne."
The Pergamum Satan Throne was the official center of emperor worship in Asia, with temples to Emperors Augustus and Trajan. Rome was Satan's capital, therefore, "Pergamum" was seen by the Christian church as the Satan throne.