I took cheese and vac packed it in September.  Migfht not be August but it was warm that trip also. We had cheese for lunch for 7 days.

Get a good hardy cheese and vac pack in sets to feed one or two days. We took a nice sharp cheddar and had some sausage some of the days as well. It was getting a little "oily" by the last day, but was fine. It was just from a good brick we bought at Cub Foods.

Gimp_of_01 buys regular Wisconsin cheese out of the "ends and pieces" bin at his local store.  These are vacuum sealed.  I have also bought those 1/2 lb bricks from Kraft.  These are hermetically sealed also.

Block cheeses such as these last quite a while, many weeks at least.  They do get soft but still are good.  I'd say that shredded cheeses would congeal into a goo-ball within a short period due to pressure and temperature.

Cheese can be frozen and does not suffer when thawed....at least the more common brick, cheddar, colby and mozzarella. Extra sharp cheddar does especially well.
For a trip in August I would start with a block of frozen cheese.
Also, individually wrapped cheese sticks will easily make it a week.

Sounds like a lot of people over-think the whole cheese thing a bit.  Just buy bricks of whatever firm cheeses you like from the grocery store, foil-wrap them if you want, stuff 'em into a ziplock, and go.

But if you are looking for specifics, here's my take:

CoJack, Colby, Monterey Jack - these are all fairly soft cheeses which will get oily and squishy easily in hot weather.  They do not grate well at all when warm.  The advantage is they taste great even when they are at "room temperature" inside of a warm pack.

Cheddar - a firmer cheese that will hold up to heat better though it can get oily.  Doesn't taste the same warm as it does cold.  Sharpness is maximized the warmer it is so if you don't like real sharp cheddar, just bring the mild stuff or don't bring it at all.  If you like really sharp cheese, you haven't tasted anything until you taste a warm extra-sharp cheddar.

Swiss - I love swiss cheese in the woods.  Much like cheddar it holds up to heat pretty well, but the flavor changes less with the temperature.  Get baby swiss to avoid the annoying gigantic holes.

Pepper Jack - See Monterey Jack  above.  Biggest difference is that all the peppers in there make it possible to grate even when warm.  Maybe "grate" is the wrong word, but it crumbles nicely into something approximating the consistency of grated cheese.

All of these cheeses keep with no problem for 10 days of blistering heat.  I do wrap them in foil, but that's it.  Even once opened, they will keep for 3-4 days just fine.