Volume dependent extension of Kerr-Newman black hole thermodynamics

We show that the Hawking–Bekenstein entropy formula is modified by a factor of $8\mathrm{/}3$ if one also considers a work term in the 1st law of thermodynamics by a pressure stemming from the Hawking radiation. We give an intuitive definition for the corresponding thermodynamical volume by the implicit definition $ϵ=M{c}^2\mathrm{/}V$, which is the average energy density of the Hawking radiation. This volume scales as $V\sim M^5$, agreeing with other suggestions. As a result the corresponding Smarr relation describes an extensive entropy and a stable effective equation of state $S(E,V)\sim E^{3\mathrm{/}4}{V}^{1\mathrm{/}4}$. These results pertain for charged and rotating Kerr-Newman black holes.