The closest thing to sport that Balamb Garden has is the training centre which is used to fight dinosaurs. At first I thought it was because SeeDs don't have time to be smurfing around with recreation given "The Mission" but they have a library and the Garden Festival, so I guess they're okay with cultural learning too.

But both Trabia and Galbadia Gardens have basketball courts, and Galbadia has a smurfing ice hockey rink, and yet Balamb seems to hate it