  Temperature is usually defined for physical systems at thermal equilibrium.
Nevertheless one may wonder if it would be possible to attribute a meaningful
notion of temperature to an arbitrary quantum state, beyond simply the thermal
(Gibbs) state. In this work, we propose such a notion of temperature
considering an operational task, inspired by the Zeroth Law of thermodynamics.
Specifically, we define two effective temperatures for quantifying the ability
of a quantum system to cool down or heat up a thermal environment. In this way
we can associate an operationally meaningful notion of temperature to any
quantum density matrix. We provide general expressions for these effective
temperatures, for both single- and many-copy systems, establishing connections
to concepts previously discussed in the literature. Finally, we consider a more
sophisticated scenario where the heat exchange between the system and the
thermal environment is assisted by a quantum reference frame. This leads to an
effect of "coherent quantum catalysis", where the use of a coherent catalyst
allows for exploiting quantum energetic coherences in the system, now leading
to much colder or hotter effective temperatures.
