Non-aged cheeses like mozzarella don't need a cheese cave to ripen, so temperature doesn't matter. You could do chevre, queso fresco, and mozzarella without worrying about temperature. For hard cheeses, it's imperative to have the correct temperature and humidity for the cheese to ripen (ferment/age).
I've seen coolers done with ice blocks and water to make cheese caves in photos on the internet. That would work if you have the patience, I think! I used a little wine cooler for mine.