GFCI outlet hot when nothing's plugged into it
As mentioned in several places, GFCIs use a little bit of power even when nothing is plugged in or energized downstream.

The heat distribution sure looks like it is coming from the GFCI electronics, but it could also be heat seeping from the other side of the wall, especially if that room has higher air pressure than this one&mdash;like if the exhaust fan were running in this room and someone took a hot shower in the other room.  Presumably there is another outlet on the other side of the wall close to this one.

You could try turning off this circuit for 15&ndash;30 minutes and see if it completely cools.  That would also rule out whether there is anything downstream causing a current to flow through it.

Or you could measure its power consumption and see if it is more than 100 mW or so.  If it is more than 1.0 watts, I'd replace it.