I've never asked if they refrigerate stuff.........
In those situations, we buy a styrofoam cooler (or bring softsided cooler of our own) and buy a bag of ice along with the groceries. Then pack the perishables on ice right at the grocery checkout. Then they store your cooler and the non-perishables right on the luggage cart with your other stuff. When we did this last January, the stryrofoam cooler was less than $3 at the grocery store we stopped at. Then we can use it all through our stay replenishing it with ice from the hotel, and just leave it when we go.