  We study quantum thermalization and thermal entanglement in the open quantum
Rabi model (QRM), in which a two-level system and a single-mode bosonic field
are coupled to either two individual heat baths or a common heat bath. By
treating the QRM as an effective multilevel system and deriving global quantum
master equations in the eigenstate representation of the QRM, we study the
physical conditions for quantum thermalization of the open QRM. It is found
that, in the individual heat-bath case, the QRM can only be thermalized when
either the two heat baths have the same temperature or the QRM is only coupled
to one of the two baths. In the common heat-bath case, differently, the QRM can
always be thermalized. We also study thermal entanglement of the QRM in both
the resonant- and non-resonant coupling cases. The logarithmic negativity for
the thermal state of the QRM is obtained in a wide parameter space, ranging
from the low- to high-temperature limits, and from the weak- to
deep-strong-coupling regimes. This work paves the way towards the study of
quantum effects in nonequilibrium ultrastrongly-coupled light-matter systems.
