  Among the study of black hole thermodynamics, topology offers a novel
approach and perspective for classifying black hole systems. In this work, we
explore the thermodynamical topology of the quantum BTZ black hole by employing
the concept of the generalized free energy. To fully characterize the
thermodynamics, we introduce two distinct topological numbers. The first one is
determined by an expression, denoted by $z$, derived from the free energy.
Although it can provide us with some local physical explanations, sufficient
physical significance still lacks from a global perspective. On the other hand,
the second topological number is based on the entropy expression of the
generalized free energy, leading to a more meaningful interpretation of its
physical implications. This result highlights the natural choice of entropy as
the domain variable for the generalized free energy. Regarding the second
topological number, our analysis reveals a topological transition that is
associated with the thermodynamical stability of the ``cold" black hole state
of the quantum BTZ black hole. Furthermore, our study suggests the existence of
topological numbers beyond the conventional values of $\pm 1,0$.
