Assuming you have disabled and or uninstalled Facebook... Have you ever dropped your device? Even if there is no sign of damage, has the device ever suffered a slight fall?
I have several XZ phones (long story why) and all of them perform absolutely perfectly... Models from Japan, South Korea, the United States and the UK, all under heavy use with perfect temperature control and very long lasting batteries.
One of my several USA devices was dropped... while there is no sign of performance slow down or even cosmetic blemishes,... Still looks new... it does heat up much more than the other devices when used outside in summer temperatures, and I assume it's because of some sort of internal damage, however so slight.
The other thought I had is perhaps do you notice this heat and battery drain while using the camera? Xperia cameras have a tendency to overheat and shut down to cool off when used for long periods of time, especially in warmer climates.
The last thought I had for you... Does your sim card have a hard time maintaining a steady and strong signal? A weak signal from a service provider in any Xperia will create heat in many cases.
