Emergence of cosmic space and horizon thermodynamics in the context of the quantum-deformed entropy

Abstract According to the quantum deformation approach to quantum gravity, the thermodynamical entropy of a quantum-deformed (q-deformed) black hole with horizon area A established by Jalalzadeh is expressed as $$S = \pi \sin \left( \frac{A}{8G\mathcal {N}} \right) /\sin \left( \frac{\pi }{2\mathcal {N}}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mo>sin</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mi>A</mml:mi> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>G</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfrac> </mml:mfenced> <mml:mo>/</mml:mo> <mml:mo>sin</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:mrow> </mml:math> , where $$\mathcal {N}=L_q^2/L_{p}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>/</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> is the q-deformation parameter, $$L_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> denotes the Planck length, and $$L_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> denotes the quantum-deformed cosmic apparent horizon distance. In this paper, assuming that the q-deformed entropy is associated with the apparent horizon of the Friedmann–Robertson–Walker (FRW) universe, we derive the modified Friedmann equation from the unified first law of thermodynamics, $$ dE = TdS + WdV $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> <mml:mi>d</mml:mi> <mml:mi>S</mml:mi> <mml:mo>+</mml:mo> <mml:mi>W</mml:mi> <mml:mi>d</mml:mi> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> . And this one obtained is in line with the modified Friedmann equation derived from the law of emergence proposed by Padmanabhan. It clearly shows the connection between the law of emergence and thermodynamics. We further investigate the constraints of entropy maximization in the framework of the q-deformed horizon entropy, and the results demonstrate the consistency of the law of emergence with the maximization of the q-deformed horizon entropy.