Heather, I would most definitely refrigerate all opened and unopened lev.
The manufacturer states that opened lev should not be refrigerated, but they are assuming that a human is using it, will use it up within 28 days or nearly will, and can afford to throw out unused insulin (because insurance is covering one cartridge per month).
All of us here, that I know of, use lev to the last drop by keeping it in the fridge. I had a cartridge that lasted 8 months opened in the fridge.
If, let's suppose, you had a high dose kitty, getting more than a couple of units, then you would probably want to warm the insulin by holding the filled syringe with your fingertip surrounding the tip where the insulin is. And if a really high dose, like some acro kitties are on 10, 15, 20 units at a time, you would not refrigerate it, but you would also be using it up in a month.
Thanks so much for the help. Have a great Memorial Day everyone!