My choice would be to add a protease inhibitor cocktail, then divide the extract into single-use aliquots, flash-freeze them, and store them at -80 or in liquid nitrogen.
As you know not all enzymes, or in fact proteins in general have the same stability, & always you can check for your enzyme by follow its activity at various time intervals i.e. measure the activity when the sample is fresh, and repeat the measurement after one week, 2 weeks & so on , after store it at room temp.,in frg. in deep freezer & in liquid nitrogen if available.
Generally it is recommended if you need to store any protein sample in deep freezer for long time, you should not store a dilute sample, you should store the protein in a concentrated solution, in presence of small amount of glycerol & if it is an enzyme in presence of its substrate.
And it is more safe to keep it in a solid form & of course if you have a lyophilizer