I would also say that the power button is broken. As a matter of fact I just tried constantly pressing my working power button on my FP1 and the phone screen flashes on/off with a delay of roughly 5 or 10 seconds (not too fast). I run kitkat so there is a black screen with white Fairphone logo alternating with a completely black screen.
Here is some more information that could help you or a repair shop to repair the phone: