How long will cayenne peppers extend shelf life of cooked ground pork?
Opinion and learned knowledge without doing searches for citations to back it up to follow:

Spices, including peppers have a history of being used to extend the life of foods including meats.  This is a false reputation and representation of that history, IMO.  The spices in reality were used in large quantities to cover up poor quality proteins and produce and hide the deficiencies and early stages of spoilage.  It hid these qualities only, it did little to nothing to reverse or slow them.  Under poor storage conditions 100 years ago a bite of pork might have turned your stomach to taste it at two days, but spice it up enough with peppers or a stout curry sauce and at four days you still might not have noticed.  This does not mean it will still not end up with you locked in the loo for a few hours or worse.

With improved meat and produce quality and storage capabilities today in most of the world (yes, really, even as much as many of us complain), this is no longer necessary. We make things spicy because we are used to the taste and like it.  But stories and traditions of it being a preservative continue.  Salt for instance is a preservative, but only in the quantities used in curing and pickling and such, not in the quantities we use to spice food.  That is for flavor, not to kill bacteria or extend the useful life of the food.