Holographic thermodynamics requires a chemical potential for color

The thermodynamic Euler equation for high-energy states of large-$N$ gauge theories is derived from the dependence extensive quantities on number colors $N$. This relates energy state to temperature, entropy, degrees freedom and its chemical potential, but not volume or pressure. In context gauge/gravity duality we show that dual generalized Smarr formula black holes in presence a negative cosmological constant. We also match fundamental variational thermodynamics first law hole mechanics, when extended include variations constant Newton's