We boondocked at the North Rim two summers ago, in August. Had our cat with us. It was never too warm in the trailer, and he stayed there just fine.
There is a Forest Service campground between the Park and Jacob Lake. Indian Crossing, or something like that. You may want to consider that campground, as it is more roomy than the National Park campground. It is closer to some of the Forest Service roads that can take you out to the rim, outside the Park. Great day trips, and you can take your dog on the walk on those trails.
From our week-long experience, I don't believe you have anything to worry about, temp-wise, about leaving your pets in the trailer during the day.
GordonThree, we take our pets along, as we are not gone 24 hours a day. We are still with them more than they are alone. Beats the heck out of putting them in a kennel (doggie prison) while we are gone.