Negative heat capacity ($dE/dT<0$)

A decrease in equilibrium temperature when the total energy increases along a specified family of equilibria. A virialized nonrelativistic self-gravitating ideal gas has $E=-U$ and $U$ proportional to its particle-weighted mean temperature, producing this behavior. It does not mean the local gas parcel has negative heat capacity at fixed volume. It occurs naturally for isolated gravitational systems rather than stable ordinary canonical equilibrium with an infinite heat bath.