As long as you keep the container COMPLETELY air tight from outside oxygen, it should be able to be rejuvenated.
The problems with paste are many. First, it's a suspension, not a solution. Gravity makes the paste separate. So if the paste has separated, perhaps all you need to do is mix it up real well to make it work, again. But mixing paste causes some waste in itself.
Also, the more volatile solvents are hard to contain, so you might have to add some more to the mix.
Also, if it is exposed to oxygen it can go bad because you are building up more oxides/salts in the paste, increasing the conductivity of the residue. The acid might not be technically "active" at 40F, but that's just a buzz word. As long as the flux part of the paste is in solution, it is active... just a lot less so than when it's at 300C.
A flux like SOLID rosin is actually inert at room temp and active at 150F, or so. But in solution in a solvent, it is active enough at room temp to dissolve copper oxides and turn green with ions, if left on copper for an hour or so.