In Modern English, the only use Colosseum sees is in describing the Roman Colosseum, or Flavian Amphitheatre. In this use, it should always be capitalized. Coliseum is the spelling for all other uses, both as a common noun and the name of other venues than the one in Rome.

col·​i·​se·​um ˌkäl-ə-ˈsē-əm : a large structure (as a stadium) for athletic contests or public entertainment Etymology from the Latin spelling used in the Middle Ages for the Colosseum, a great outdoor arena in ancient Rome

colosseum noun [ C ] (also Colosseum) uk / ˌkɒl.əˈsiː.əm / us / ˌkɑː.ləˈsiː.əm /

Colosseum, the From Longman Dictionary of Contemporary English Col‧os‧se‧um, the /ˌkɒləˈsiːəm $ ˌkɑːl-/ (also the Coliseum) an amphitheatre (=large circular theatre without a roof) in Rome which was built during the first century AD and is now one of the most famous ancient Roman buildings.