CLIMATE MQTT availability

A “Climate” entity that is created by MQTT autodiscovery remains “available” even if actual device stopped publishing MQTT hours ago.
With MQTT sensor entities one can add “expire_after” to the config JSON message.

Question: is it possible to make the MQTT Climate entity go into unavailable when the actual device becomes silent/stale after a time period ? (typically add “expire_after” to the CLIMATE MQTT config ? )

all suggestions wellcome please.

Did you manage to solve it? I faced the same situation