It’s medium heat, but also most modern dryers have a “cool down” cycle at the end of the permanent press cycle. If hot permanent press clothes sit in a pile after the dryer stops they are more likely to wrinkle (the name “permanent press” notwithstanding). So the cool down helps prevent wrinkles.
The larger question remains: Can I get clothes which have been treated for permanent press re-treated? Because it seems like it doesn’t last very long.
You can buy both spray starch and spray sizing, which will last for one washing cycle (they aren’t water-permanent) if you want clothes to be less wrinkly. You can also buy “wrinkle release” which is basically the same deal, but in a format that is usually detergent-and-water-resistant (not proof) and will last for about 6 to 20 washing cycles before wearing out. If you don’t want to bother with that, and want something more permanent, you’re mostly SOL.
This is a blog which, while obviously anti-chemical, has the basics of modern vapor phase formaldehyde “permanent press” treatments down pretty clearly. Despite the tone, the actual information matches up pretty well with what I learned in my fabrics classes. Basically it’s a chemical and temperature-extreme process that is most easily done in large batches in factories designed for the purpose.
Most large dry-cleaners will offer a “re-permanent-press” service, but basically, they’re just charging you lots of money for a slightly different formulation of spray starch, spray sizing, or spray “wrinkle release” that will last perhaps half again as long as the stuff you can buy from the store yourself. In that case, you’re mostly paying for the convenience of not having to hang and spray down your clothes yourself.