This paper explores the thermodynamical topology of the quantum BTZ black hole using the concept of generalized free energy. The authors introduce two distinct topological numbers to characterize the thermodynamics of the quantum BTZ black hole. The first topological number, derived from the free energy, provides local physical explanations but lacks global significance. The second topological number, based on the entropy expression of the generalized free energy, offers a more meaningful interpretation of its physical implications. The study reveals a topological transition associated with the thermodynamic stability of the "cold" black hole state of the quantum BTZ black hole and suggests the existence of topological numbers beyond the conventional values of \(\pm 1\) and 0. The paper also discusses the critical boundary obtained from \(\partial_z S = 0\) and the vector mapping, which helps in understanding the thermodynamical topology. Through comparative examples, the authors illustrate the physical interpretations of the local topological numbers and the global topological number, highlighting the importance of entropy as the domain variable for the generalized free energy. The results provide insights into the thermodynamics and topology of the quantum BTZ black hole, emphasizing the significance of topology in this field.This paper explores the thermodynamical topology of the quantum BTZ black hole using the concept of generalized free energy. The authors introduce two distinct topological numbers to characterize the thermodynamics of the quantum BTZ black hole. The first topological number, derived from the free energy, provides local physical explanations but lacks global significance. The second topological number, based on the entropy expression of the generalized free energy, offers a more meaningful interpretation of its physical implications. The study reveals a topological transition associated with the thermodynamic stability of the "cold" black hole state of the quantum BTZ black hole and suggests the existence of topological numbers beyond the conventional values of \(\pm 1\) and 0. The paper also discusses the critical boundary obtained from \(\partial_z S = 0\) and the vector mapping, which helps in understanding the thermodynamical topology. Through comparative examples, the authors illustrate the physical interpretations of the local topological numbers and the global topological number, highlighting the importance of entropy as the domain variable for the generalized free energy. The results provide insights into the thermodynamics and topology of the quantum BTZ black hole, emphasizing the significance of topology in this field.