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

Abstract 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 cast into four universal thermodynamic classes, denoted by $$W^{1-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:msup> </mml:math> , $$W^{0+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> , $$W^{0-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:msup> </mml:math> , and $$W^{1+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> . Our study reveals that the Schwarzschild black hole in PFDM belongs to the $$W^{1-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:msup> </mml:math> class, and the independence of black hole size is thermodynamically unstable in both the low- and high-temperature limits. The Reissner–Nordström, Kerr, and Kerr–Newman black holes in the PFDM background belong to the same universal thermodynamic class, $$W^{0+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> , 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 when compared to their counterparts in PFDM. The Schwarzschild–AdS black hole belongs to the $$W^{0-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:msup> </mml:math> class, indicating that no black hole state exists at low temperatures, but small-unstable and large-stable black hole states exist at high temperatures. Furthermore, the Kerr–AdS black hole belongs to the $$W^{1+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> 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. Thus, the presence of PFDM does not affect the stability of the black hole. These findings uncover the universal topological classifications underlying black hole thermodynamics, offering profound insights into the fundamental principles of quantum gravity.