Non-extensive entropy and holographic thermodynamics: topological insights

Abstract In this paper, we delve into the thermodynamic topology of AdS Einstein–Gauss–Bonnet black holes, employing non-extensive entropy formulations such as Barrow, Rényi, and Sharma–Mittal entropy within two distinct frameworks: bulk boundary and restricted phase space (RPS) thermodynamics. Our findings reveal that in the bulk boundary framework, the topological charges, influenced by the free parameters and the Barrow non-extensive parameter $$(\delta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , exhibit significant variability. Specifically, we identify three topological charges $$(\omega = +1, -1, +1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . When the parameter $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> increases to 0.9, the classification changes, resulting in two topological charges $$(\omega = +1, -1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . When $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> is set to zero, the equations reduce to the Bekenstein–Hawking entropy structure, yielding consistent results with three topological charges. Additionally, setting the non-extensive parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> in Rényi entropy to zero increases the number of topological charges, but the total topological charge remains (W = +1). The presence of the Rényi non-extensive parameter alters the topological behavior compared to the Bekenstein–Hawking entropy. Sharma–Mittal entropy shows different classifications and the various numbers of topological charges influenced by the non-extensive parameters $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . When $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> have values close to each other, three topological charges with a total topological charge $$(W = +1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are observed. Varying one parameter while keeping the other constant significantly changes the topological classification and number of topological charges. In contrast, the RPS framework demonstrates remarkable consistency in topological behavior. Under all conditions and for all free parameters, the topological charge remains $$(\omega = +1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with the total topological charge $$(W = +1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This uniformity persists even when reduced to Bekenstein–Hawking entropy, suggesting that the RPS framework provides a stable environment for studying black hole thermodynamics across different entropy models. These findings underscore the importance of considering various entropy formulations and frameworks to gain a comprehensive understanding of black hole thermodynamics.