In popular culture Christians were being killed in the Colosseum because they did not want to accept the Emperor as a god. An example can be the novel Quo Vadis by Henryk Sienkiewicz, who won the Nobel Prize for this. The Colosseum is a Catholic sacred place, however, not because it was used to kill Christians, but because many people died there.
The Jews for example had never accepted Caesar to be a god and nothing happened. The Greeks had their own Pantheon, very similar but a little different.
How (in reality) did the Roman Empire treat people who did not want to accept Roman gods? Were they killed or punished (eg. with a fine)? Or was it just not important to anyone?
In Wikipedia an article (with a "citation needed" remark) states:
In western Classical Antiquity, theism was the fundamental belief that supported the divine right of the state (Polis, later the Roman Empire). Historically, any person who did not believe in any deity supported by the state was fair game to accusations of atheism, a capital crime. (...). Christians in Rome were also considered subversive to the state religion and persecuted as atheists
(but maybe atheism was something worse than just believing in another religion)