  Entanglement entropy in conformal field theories is known to satisfy a first
law. For spherical entangling surfaces, this has been shown to follow via the
AdS/CFT correspondence and the holographic prescription for entanglement
entropy from the bulk first law for Killing horizons. The bulk first law can be
extended to include variations in the cosmological constant $\Lambda$, which we
established in earlier work. Here we show that this implies an extension of the
boundary first law to include varying the number of degrees of freedom of the
boundary CFT. The thermodynamic potential conjugate to $\Lambda$ in the bulk is
called the thermodynamic volume and has a simple geometric formula. In the
boundary first law it plays the role of a chemical potential. For the bulk
minimal surface $\Sigma$ corresponding to a boundary sphere, the thermodynamic
volume is found to be proportional to the area of $\Sigma$, in agreement with
the variation of the known result for entanglement entropy of spheres. The
dependence of the CFT chemical potential on the entanglement entropy and number
of degrees of freedom is similar to how the thermodynamic chemical potential of
an ideal gas depends on entropy and particle number.
