Any hard cheese is fine. I suspect most people take cheddar, but there are many other hard cheeses. Just get the real thing, not "process" cheese. Soft cheeses, such as brie or mozzarella, of course spoil much faster and should be used within a couple of days. While all cheeses are now sold refrigerated, that's only because they keep longer that way.

Wrapping the cut cheese in cheesecloth, so the cut surface dries out, makes it last much longer. If there's mold, it will be only on the surface and can be trimmed off and discarded. Most good cheese "bricks" are still coated with wax as they were back in the old days. Avoid plastic because it seals in the moisture and contributes to mold or worse, slime (even in the refrigerator).

My mother used to buy a "brick" of sharp cheddar, cut it in thirds, wrapped it in cheesecloth and then in wax paper. Two of the thirds were mailed ahead with our resupplies so sat from 2 to 4 weeks in a warm post office. She very seldom had to trim off any mold during a six weeks' trip.