Adams peanut butter can safely stay out of the refrigerator and if you leave the Adams peanut butter out of the refrigerator it will taste better.
I've never refrigerated any brands of peanut butter including Adams peanut butter because it really doesn't need it and if you open the peanut it will last for a month or more before it goes bad.
The peanut butter in this house though hardly lasts a month because we love peanut butter and eat it a lot so I'm constantly buying more peanut butter.
So it never really has time to sit and go bad.
You can refrigerate the Adams peanut butter or any other peanut butter if you desire but there really is no reason to do so.