We show that the Hawking–Bekenstein entropy formula is modified by a factor of 8/3 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=Mc2/V, which is the average energy density of the Hawking radiation.

This volume scales as V∼M5, 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)∼E3/4V1/4. These results pertain for charged and rotating Kerr-Newman black holes._x000D_

T.S. Biró, V. G. Czinner, H. Iguchi, P. Ván, Volume dependent extension of Kerr-Newman black hole thermodynamics, Physics Letters B 803(10) (2020) 135344