My PDX10 stays on continuously and so drains the battery if left on "Camera" for extended periods.
My previous camera was the V900 which went into "Sleep" mode after 5 minutes but could be 'woken up' by pressing the "Stop/Start" button.
I have scoured the manual and the menus on the camera but have been unable to see how to activate an "Auto sleep" mode after 5 minutes of inactivity.
Any help would be appreciated as I am using the PDX10 remotely without physical access to the "On/Off" switch.

I have the same experience and question. Looking forward to hearing the solution, if there is one.

I have drained several batteries by accidently switching the camera to on while storing it in my dry box.

I have a cheap Sony tripod that has a LANC remote control that can power the camera on and of, control zoom and recording. It uses a cable that I think could be very easily extended.

I try to store the camera without the battery lately to keep that other problem from happening.

It does power the head down after 5 mins though doesn't it? I assume that would save most of the power being used.

Unfortunately not. The mono viewfinder stays on all the time and the battery runs down almost in real time.
I have actually solved the problem by fitting a mechanical on/off switch to the underwater housing in which i use the PDX10.
Many thanks for your input