Universal topological classes of black holes surrounded by quintessence

Abstract This work explores the universal topological classes of various black hole configurations immersed in a quintessence field. The results indicate that the Schwarzschild black hole surrounded by quintessence is of the $$W^{1-}$$ W 1 - type, exhibiting thermodynamic instability in both extremal temperature limits. The introduction of rotation alters its topological nature to the $$W^{0+}$$ W 0 + class, where the Kerr black hole displays coexistence of a stable small black hole and an unstable large black hole at low temperatures. In comparison, anti de Sitter (adS) black holes display different thermodynamic topologies. The Schwarzschild–adS solution corresponds to the $$W^{0-}$$ W 0 - class, where an unstable small branch and a stable large branch emerge in the high temperatures. Meanwhile, the Kerr–adS classified as $$W^{1+},$$ W 1 + , maintains stability in both high- and low-temperature limits. Furthermore, the 4D/5D quintessential black hole belongs to the $$W^{0+}$$ W 0 + class within the Einstein–Gauss–Bonnet framework. Overall, parameters such as rotation and Gauss–Bonnet in quintessence background field affect the topological classifications of the black holes, while the quintessence field primarily modifies the stability range without changing the topological class itself.