Thermodynamic phase transition and winding number for the third-order Lovelock black hole

Phase transition is important for understanding the nature and evolution of the black hole thermodynamic system. In this study, we predicted the phase transition of the third-order Lovelock black hole using the winding numbers in complex analysis, and qualitatively validated this prediction by the generalized free energy. For the 7<d<12-dimensional black holes in hyperbolic topology and the 7-dimensional black hole in spherical topology, the winding number obtained is three, which indicates that the system undergoes first-order and second-order phase transitions. For the 7<d<12-dimensional black holes in spherical topology, the winding number is four, and two scenarios of phase transitions exist, one involving a purely second-order phase transition and the other involving simultaneous first-order and second-order phase transitions. This result further deepens the research on black hole phase transitions using the complex analysis