The Disjointed Thermodynamics of Rotating Black Holes With a NUT Twist

We study the solutions of the Euclidean-signature Einstein equations (gravitational instantons) whose line-element takes the Kerr-bolt form, characterized by three real parameters: a size, a NUT charge, and a spin rate. The exclusion of singularities eliminates most combinations of these parameters, leaving only separated solution-manifolds between which continuous transitions are impossible (The angular velocity divided by the temperature is forced to be a rational multiple of $2\pi$). This ``quantization'' prevents the free variations presupposed by an equation like $T dS=dM + \Omega dJ$, and thereby renders the first law true, false, meaningless, or tautological, depending on how one approaches it