If you can’t wait for Littlebigplanet Vita to release, then you might want to head over to the Japanese PSN store, as the demo has released over there. The demo is 986 MB, which means it’s most likely the same version as the preview, since they’re quite close in file size.
For those living in the US, you can download it on September 25, which is also when it releases. However you can check out our hands on preview in the mean time. Finally, make sure to check back closer to the release, for our review of Littlebigplanet Vita.