Right after a update of Ubuntu Server I faced that rsyslog used 100% CPU and that /var/log/syslog got filled up with following line, which was repeated almost every second:
[rsyslog] imklog: error reading kernel log - shutting down: Bad file descriptor
The problem is, that rsyslog tries to log kernel messages, which is as OpenVZ/Proxmox Container default not possible - to fix this, remove/comment out following line in the file /etc/rsyslog.conf (here already done):
# $ModLoad imklog # provides kernel logging support
After that, restart rsyslog and the problem should be gone.