My other, small quad copter does what your Phantom is doing when the batteries reach about 40 % also.
It just seems to lose it's motivation.
The Phantom is vastly more complex.
But the cause should still be the same....less current reaching the motors for whatever reason.
That's the only way you lift off and hover. The controller allows more current to flow to the motors than at idle. More current = more torque, higher RPM, more lift.
If only you could determine exactly what's reducing the current, you'd solve the mystery (right?)
Is it the battery? The Controller board? The Naza? You need that data diagnostic tool ianwood has.
But one thing....the fact that you lose connection to the app.... maybe the Phantom is going into auto land because it knows battery power is low and it can't RTH due to low power....so it just reduces power to land? Grabbing straws. I'm sure that's not it....just throwing stuff out.