Upvote:3
You will not need to go through security again, but you will have to go through passport control. Since Svalbard is governed under the Svalbard Treaty, flights are from the international terminal, and since you're coming from FRA, you will have already gone through immigration to get into the Schengen area. This is not an issue though, as there are plenty of good places to sleep in the domestic section of OSL. Passport control opens at around 5:30am.
There are a few potential exceptions to the above: