Holographic equipartition and thermodynamic evolution of topological dilaton black holes with power-law electrodynamics

Abstract We investigate the thermodynamic structure of higher-dimensional topological black holes in Einstein–dilaton gravity coupled to a power-law nonlinear Maxwell field. While the geometric and equilibrium thermodynamic properties of these solutions are known, their interpretation through holographic equipartition remains unexplored. Building on recent work in scalar–tensor gravity, we show that spacetime evolution can be understood as the system’s tendency to restore balance between bulk and surface degrees of freedom—provided the thermodynamically consistent Jordan-frame action (augmented by a $$\Box \phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>□</mml:mo> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:math> boundary term) is used. In this framework, the entropy density acquires a dilaton correction $$\propto \phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∝</mml:mo> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:math> , and the local Unruh temperature is governed by a modified acceleration incorporating scalar–field gradients. The natural thermodynamic surface coincides with $$N\sqrt{\phi } = \text {const}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:msqrt> <mml:mi>ϕ</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mtext>const</mml:mtext> </mml:mrow> </mml:math> —precisely the radius employed in Brown–York mass computations. We compute the departure from holographic equipartition and find that increasing the dilaton coupling $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> or the electromagnetic nonlinearity p enhances disequilibrium, especially near extremality. All results recover the RN-AdS limit when $$\alpha = \gamma = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$p = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Finally, we offer a thermodynamic interpretation of the observed dependence of quasinormal mode damping on $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and p , based on the radial sensitivity of the effective $$T'_+ S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> product. This is consistent with the idea that geometric deformation imprints itself not only on the background geometry, but also on dynamical perturbations—through their thermodynamic response.