I have seen a few people who keep saying it HAS to be spelt Goukaiger because Gokai with short O means misleading, misunderstanding (誤解) or it is a Buddhist concept (五戒). But, Goukai with a long O means thrilling or exciting and that's the word they are trying to use. I don't think it matters, the show is actually called 海賊戦隊ゴーカイジャー, so anything else is just a different way to spell it

I don't have a favourite Gokaiger yet so I am using your icon of the group instead