It's a national park and there are roads and ranger stations and such, so it doesn't feel very wild or remote. .
Incidentally, what are some strategies to stay cool in a broken down car in a desert? Cover the car with tarp or mylar, open the doors, have ice in portable freezer. Any other ideas?
While most national parks do have road access, they contain plenty of raw, screaming wilderness and many locales that are quite remote. There are large areas of some Alaskan parks that essentially have never been visited, by rangers or anyone else.
Keeping cool in desert conditions is a matter of creating shade and opening up to breezes as much as possible. The classic structure is a ramada - think a roof supported on poles - open on all sides to the open air. I would get out of the car and use my tarp to cover as wide an area as possible, using brush to support the covering. Expend as little effort as possible in doing this.
Rather than stay in the car, I would use the car to support part of my ramada.
Another option is seeking the shade of an arroyo bank, rock shelter, or anything else that offers shade. Arroyos will also support vegetation and possibly moist soil or water pools that would aid considerably.
Whatever you do you don't want to walking in the sun at midday, even with a hat with a generous full brim. Wait until sundown to travel. The best hours for hiking are usually from midnight to around 8 AM - then hole up.