Depending on the local language/locale code settings of your Linux-computer, it is likely that GRID Autosport doesn't fully load.
When you launch it, the pre-game options-panel displays, and after closing it, the GRID Autosport Logo is displayed, but the game loads forever.
There is a very easy fix for this problem:
Set the launch-options for GRID Autosport in your Steam-Client to
and GRID Autosport will work fine!
Source: Steam Forum