Hacker Marcan of team Fail0verflow has been working on Linux for PS4 for more than a year, and demonstrated it running on firmware 4.05 along with 3D acceleration, in December during the 33C3 convention in Germany. A couple days ago he twitted that he is getting close on getting Vulkan to work on the PS4.
What is Vulkan?
Vulkan is a set of APIs for graphics rendering, a bit like DirectX or openGL. Many actually see it as the spiritual successor of openGL. For the end user, what this typically means is that in the long term, games and applications using Vulkan will have better graphics and better rendering. Graphics hardware has been evolving at a crazy pace, and it needed open source graphics APIs to follow. That’s pretty much where Vulkan comes in play.
Marcan has been working on porting the AMDGPU open source graphics drivers from AMD onto Linux for PS4. I assume Vulkan compatibility will be an intentional side effect of this port.