The entrance to the Unstable Hall of Knowledge is located at the Lake of Solitude, where the 701st Garrison is also situated.

The entrance opens every few hours and disappears again shortly after.

You need the [plur]Garrison Crystals from the 701st Garrison and 712th Garrison to craft the key you need to enter.

To find out more, you need to find the scout stationed at the Lake of Solitude. Take the Red Crystal Sample Key[pl:"Keys"] to Palarus and find out what the latest news is.
The research on the Red Crystal Key seems to be complete.

I also found out something about the opening times for the entrance to the Unstable Hall of Knowledge.

Apparently, the entrance opens every two hours.

Then it remains open for 15 minutes.

That's all I know so far. Please bring my findings to the Tribune.
Now we know when and for how long the entrance opens.

That means we're inching closer towards solving this puzzle.

You should enter the Unstable Hall of Knowledge when you get a chance, Player.