Here are your parts
Replacement parts for MAYTAG MDE4000AYW DRYER- ELE | AppliancePartsPros.com

Here is a service manual
16023110 Maytag Sloped Front Dryer Repair Service Manual - ApplianceDigest.com
You will have to join the site to download the manual but it is a very good site for info.

First I would check that the heating coil will shut off in an air only cycle.
If not you may have a grounded element.
Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to be on and the thermostats/switches cannot regulate it.

[COLOR="DarkRed"]The dryer drys fine, the only problem I have is with the cool down cycle do I need to check the heating coil?[/COLOR]
Yes because if it is grounded it can still heat OK, depends on where and how it grounds. But it will not shut off and that is required during the permanent press cool down.

OK.
Are you sure that the heater is staying on for the whole cycle?
This is something I am sure you have already checked but it never hurts to double check things.

It looks like the heater gets power through the timer contact set 3.
If you look at the timing chart page 10-3 in the manual they open before the drive motor contact set 1.

Unplug the unit, remove one wire/side of contact set 3 and measure across the contacts.
Manually move the dial to see where/if they open.

You are welcome

And thanks for getting back to us. Now when others search for a similar problem they will see what actually worked instead of just suggestions about what could be the cause.