After installing Mint 13 I noticed that there is no Audio-Output for my TV which is connected via HDMI-Cable.

I searched and tried many different solutions until i had enough knowledge to search for the right term :-)

it is: "pulse audio add hdmi"

https://wiki.archlinux.org/index.php/PulseAudio/Examples#Finding_HDMI_output

To setup which app uses which audio output get the package pavucontrol.