It's weird that you're seeing such high CPU usage with the open source drivers. AMD has been directly supporting them for a while now. For most applications radeonsi (the open source driver for GCN arhictecture cards) performs within spitting distance of the proprietary driver and for some the performance is better. I suppose it's possible they are more CPU intensive generally, but the kind of slowdown you're seeing seems pretty excessive. What applications are you seeing this problem with?
Ubuntu 16.04 does ship with a pretty old version of Mesa (full major version behind, about to be 2 major verions as Mesa 13 is in the RC stage) which probably isn't helping matters. Unfortunately, while there are some PPAs that make it fairly easy to install a bleeding edge build, there's not really a convenient way to install the latest stable release.
For some context, the reason fglrx/Catalyst has not been updated is due to an in progress driver transition for AMD cards on Linux. Generally speaking, Linux video drivers are split into a kernel part and a user space part. For quite some time, AMD has maintained two completely separate driver stacks on Linux. For the user space side, this didn't represent a huge amount of duplication of effort as most of the code is shared with the OpenGL portion of their Windows driver, but for the kernel side it was a bunch of wasted effort. A while back, they started on a new open source kernel driver, called amdgpu, that could provide the necessary facilities for both their open source and proprietary driver efforts. The new proprietary driver (AMDGPU-Pro) targets this kernel module. Unfortunately, amdgpu does not have production-ready support for GCN 1.0 GPUs like the 8570. This will get fixed eventually, but that doesn't really help you now.
