There is no specific “Librem One” application for Linux. I find it better to think of Librem One as being access to services, and how I connect to the services is up to me.
For instance, you get an email address. Whether you use Thunderbird or Geary or any other email client is your choice. You also get a Matrix account. You can use Riot or Fractal or whatever else. You can use your Librem Social account in a browser.
Purism does a lot of working working upstream with GNOME, so most if you want the “Librem One app” on Linux, it would be the default GNOME app. Geary will probably become the default email client, Fractal may be the default Matrix client, and whileI don’t use the Librem Social stuff really, so I know less about that, it looks like there is a program called Tootle that would be a Mastodon client.