Latest PHP FPM on Ubuntu

Linux vps-XXXXXX 6.2.0-39-generic #40-Ubuntu SMP PREEMPT_DYNAMIC Tue Nov 14 14:18:00 UTC 2023 x86_64 x86_64 x86_64 GNU/Linux

Using

sudo add-apt-repository universe
sudo apt install php-fpm php-mysql

And we got

php8.1-fpm

service php8.1-fpm status

You nginx confs should be also /run/php/php-fpm.sock