A homebrew enabler has been released for PS4 consoles on Firmware 4.05 by VVildCard777. The PS4HEN homebrew enabler allows PS4 users on firmware 4.05 to not only launch homebrew applications, but also make use of custom themes and pkg's. As with firmware 1.76 booting a fully working Linux distribution with 3d acceleration is also possible, thanks to the previous work by fail0verflow.
PS4HEN is currently available on VV1LD's GitHub page, with instructions on how to make use of it. Users must first run the 4.05 kernel exploit and inject the payload that way, after that things are as simple as installing homebrew pkg's through Settings > Debug Settings > Game > Package Installer and running it the same as any other game installed on the PS4.
We will likely be updating the Digiex test PS4 from 1.76 to 4.05 to bring some new tutorials for 4.05 based firmware in the near future. Until then you would be wise reading up on how you can block PS4 System Updates and for the more adventurous, exploring VV1LD's GitHub page if you wish to get started right away.
The same principles as our old PS4 1.76 guides apply, so you could read up on our Digiex's PS4 Homebrew Guides to start learning more right now.