Thermodynamics for higher dimensional rotating black holes with variable Newton constant *

The extensivity for the thermodynamics of general $D$-dimensional rotating black holes with or without a cosmological constant can be proved analytically, provided effective number microscopic degrees freedom and chemical potential are given respectively as $N=L^{D-2}/G, \mu= GTI_D/L^{D-2}$, where $G$ is variable Newton constant, $I_D$ Euclidean action $L$ length scale. In cases i.e. Myers-Perry holes, physical mass intensive variables expressed explicit macro state functions in extensive simple compact form, which allows an analytical calculation heat capacity. results indicate that zero, one $k$ equal rotation parameters all thermodynamically unstable.