Restricted phase space thermodynamics of 4D dyonic AdS black holes: insights from Kaniadakis statistics and emergence of superfluid $$\lambda $$-phase transition

Abstract We study the thermodynamics of 4 D dyonic AdS black hole in the Kaniadakis statistics framework using the Restricted Phase Space (RPST) formalism. This framework provides a non-extensive extension of classical statistical mechanics, drawing inspiration from relativistic symmetries and presenting a fresh perspective on black hole thermodynamics. Our study analyzes how including Kaniadakis entropy modifies the phase transition of the dyonic black holes. We consider the central charge C and its conjugate chemical potential $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> as the thermodynamic variable along with others except the pressure and volume. Due to the addition of the magnetic charge $$\tilde{Q}_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>m</mml:mi> </mml:msub> </mml:math> , the study of the phase transition becomes much richer by obtaining a non-equilibrium phase transition from an unstable small black hole to a stable large black hole along with the Van der Waals and superfluid phase transition with an extra unstable branch in the $$T-S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>-</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> processes. In the $$F-T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>-</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> plot, we get an extra Davies type with an extra branch phase transition. Including the deformation parameter $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> introduces an unstable (ultra-large BH) branch seen in almost all the plots. Turning off the magnetic charge flips the direction of the phase transition seen during its presence. Also, in the plots varying $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> match with the plot varying C which underlines some sort of correspondence in its meaning which is not possible to observe in Gibbs–Boltzmann statistics. As the entropy models change the homogeneity is not lost where mass is of the first order and the rest is zeroth order. The $$\mu -C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> processes in quite similar across black hole systems and entropy formulation marking some kind of universality of this process. Our study shows that modified entropy, unlike in certain alternative gravity models, does not give rise to a new thermodynamic universality class but retains consistency with standard Einstein–Hilbert black hole behavior.