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

LANG=C %command%

and GRID Autosport will work fine!

Source: Steam Forum