Universal thermodynamic topological classes of black holes in perfect fluid dark matter background

In this paper, we study the universal thermodynamic topological classes of a family of black holes in a perfect fluid dark matter (PFDM) background. Recent research on black hole thermodynamics suggests that all black holes can be classified into four universal thermodynamic classes, denoted by $W^{1-}$, $W^{0+}$, $W^{0-}$, and $W^{1+}$. Our study reveals that the Schwarzschild black hole in PFDM belongs to the $W^{1-}$ class, and independence of black hole size thermodynamically unstable at both low- and high-temperature limits. The Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes in the PFDM background belong to the same universal thermodynamic class, $W^{0+}$, which represents small, stable black holes and large, unstable black holes at low-temperature limits, whereas no black hole state exists at high temperatures. The AdS black holes behave differently compared to their counterparts in PFDM. The Schwarzschild-AdS black hole belongs to the $W^{0-}$ class, indicating no black hole state at low temperatures, but small, unstable and large, stable black hole states at high temperatures. Furthermore, the Kerr-AdS black hole belongs to the $W^{1+}$ class, characterized by small, stable black holes at low temperatures, large, stable black holes at high temperatures, and unstable, intermediate-sized black holes at both low and high temperatures. These findings uncover the universal topological classifications underlying black hole thermodynamics, offering profound insights into the fundamental principles of quantum gravity.