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.