About 30 years ago we had a polystyrene box (I appreciate that this is not the ideal material, but there is a thread somewhere else here relating to uses for such stuff)with a shallow depression on the top. Whilst not full refrigeration it did work as an efficient coolbox.
I let it grow where it wants, cos it’s gorgeous statuesque and the insects love it, remove it if it’s too obstructive , but otherwise let it grow, why not!