Hm, Pathfinder Adventure Path #9 introduced us to the fine sport of blood pig. That's the only original sport that comes to mind at the moment, but you could probably come up with some fun ideas by browsing wikipedia's list of team sports (surely they'd have some form of pato, for example...).
Andoran has a couple: Prismati and the annual Carpenden food fight.
One of the Kingmaker adventures features a few, jousting, insult-slanging, and log-cutting if I remember correctly. Might have been some drunk archery involved as well.