I have an older model whirlpool oven, approximately 10 to 13 yrs old. Last night at some point the oven heated up randomly we found it set 345* this morning. What could cause this?

Oven or stove comes on by itself:
The broiler - range broiler comes on by itself and stays on until power cut. The control board is likely faulty or the broil element filament shorted to the broil element outer housing. If the broil element looks normal, no cracks or blistering, I would suspect a faulty electronic oven control.

The only other possibility is an unlikely one but you could have a short in the wire harness which could allow current to flow through at LEAST one leg on the power supply through the element uncommanded. Stuck keys are very common on some ovens and RARELY are fixed without replacement of the control. (but can try cleaning them)

Hint hint is the oven is programmable, and someone might have actually programmed it to come on without realizing it or a power surge reset it!
Read more:http://www.justanswer.com/appliance/2docw-amana-range-model-xxxxx-broiler-turns-itself.html#ixzz3YdTBmqIr

BURNERS: the burners going on BY themselves, since there are usually a number of safeguards involved. Such as the ignition spark IS only produced when the knob has been pushed in and rotated to that position. Not to say that it can't happen ... Which if it does I would suspect the wire harness or infinite switch is shorted somewhere?

GAS oven starts on its own: Possibly a contact inside the thermostat is telling the burner to come on sporadically causing the oven to heat up.
The thermostat CONTROL should prevent the igniter from getting voltage when itis in the "off" position. Hence a defective thermostat may cause ignition on its own. The thermostat is the part behind the turn knob that you turn on the oven with. TO TEST OVEN CONTROL:Disconnect one wire from the terminals being tested, and clip on the METER tester probes. Set the oven temperature dial to 300°F. If the control has more than two terminals, identify which terminals to test, using the diagram located on the rear panel, inside the storage drawer or control panel, or in the owner's manual.
Test confirm with a multi meter for OHMS continuity when in the on and offposition. If it stays on or closed in the off position its defective.
1.To remove:capillary tube:Capillary tube is a rod like thing that extends about 3-4 inches from the back of the oven. If you look inside the oven, you will be able to find the capillary tube. The capillary tube senses temperature inside the oven, this information is passed through the rod to the temperature control. When the temperature of the oven reaches the set temperature at the temperature control, the power is turned OFF. When the temperature goes down, the power is turned on and the oven heats up again .The capillary tube is generally held by screws and a cover plate.
2.Remove capillary tube:If your range is self-cleaning type, you need to take extra care. The capillary tube in a self cleaning range normally has toxic liquid; the liquid may spread into your body. Be ready with cold soapy water and a towel. If liquid falls on you by accident, carefully wipe off the liquid with the towel and wash the affected area with soapy water.
Remove the screw and the cover plate that hold the capillary tube (if there is any). Push it slowly through the hole at the back of the oven. Be careful not to strain the rod that goes to the temperature control. Pull out the capillary tube from the oven.
Remove temperature control:Label the wires connected to the temperature control. If possible, take a close-up picture of the terminals and the wires using a digital camera.

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The reason for my free advice is GOD is good!