Jun 10, I am wondering if there other cheeses that don't need to be or shouldn't be refrigerated? I'm trying to make a gift basket and I want to include. Some types of cheese require constant refrigeration and others do not. In general , cheese with high moisture content, such as ricotta or mozzarella, should not. Jun 4, "With soft cheese, you have to worry about the texture and how it's going to hold up. You also don't want to worry about something that will start.