My truck has recently started doing the high-idle after EOC. I haven't tried varying the key-off time to see if it has an effect, but I have to generally keep the key off for 2-3 seconds to keep the engine from restarting on its own.
I have noticed that my factory original battery (in a truck manufactured Feb 2002, purchased Aug 2002!) is holding less of a charge. I plan to replace it this winter or it will surely die when it gets really cold. When I am in EOC I notice my voltage gauge dipping halfway between nominal and the "low" tick mark. I can no longer EOC at night or my headlights dim far too much far too quickly.
A lot of vehicles have an automatic high idle to bring alternator speed up and improve battery state of charge. If your ECU monitors battery voltage it could be trying to recharge after the bump start. Why this would be changed with a second key-off cycle I don't know, but it is another possibility.
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