Question

how was the colosseum paid for

Answer 1

"Just prior to the construction of the Colosseum, the newly crowned Emperor Vespasian commissioned his son, Titus, to squelch the Jewish rebellion that was growing in Judea c.68/69 AD. By 70 AD Titus marched confidently into Jerusalem with the Roman Twelfth Legion. This Roman force conquered Jerusalem and slaughtered 1.1 million people according to Josephus, and destroyed much of the city and Temple along with it."
The Roman colosseum was financed by the stolen treasure from the people of Jerusalem.

Answer 2

No one knows exactly how much the building of the Colosseum cost. But in A.D. 70 Titus had sacked the city of Jerusalem. The fabulous treasure of Jerusalem paid for the building of the Colosseum, and no expense was spared in the project.