As you would expect with HP thin clients with a PC compatible architecture and standard BIOS there should be no problem running various flavours of Linux on the t510.
I tried it out with Tiny Core 6.1 which was installed on an 8GB pen drive.
In May 2022 I heard from GJ, an Asterix user:
...and successfully installed AstLinux on it (latest version with Asterisk v18.10). It runs fine, I didn't encounter any problems and installation was very straightforward as described on the AstLinux website.