There's two noteworthy aspects:

Path Adjustments in libvirt / qemu

I'm running Home Assistant OS as a virtual machine on my home server. The HASS OS VM image relies on an OVMF file for booting; this file moved to a different path.

So, in order to make Home Assistant boot again after the upgrade to Debian 13, edit /etc/libvirt/qemu/hassos.xml (or similar) and change /usr/share/OVMF/OVMF_CODE.fd to /usr/share/OVMF/OVMF_CODE_4M.fd.

paho.mqtt.client changes

The upgrade from python3-paho-mqtt 1.6.1 to 2.1.0 came with some changes in its API. In order to make it happy (and not spew any warnings), replace paho.mqtt.client.Client() with paho.mqtt.client.Client(callback_api_version=paho.mqtt.client.CallbackAPIVersion.VERSION2).

Apart from that, everything went smoothly.