I must admit that I am not that familiar with this topic. I've paid attention to the research surrounding materials with negative index of refraction, or right-handed materials, that have been made recently using metamaterials. However, I have not paid any attention to materials having a negative specific heat.
It appears that such material can be shown to violate the Zeroth Law of Thermodynamics, as shown in this preprint that is to be published in PRL. Obviously, systems that exhibit such specific heat are not something one encounters everyday (are these systems even under what we generally consider to be a thermodynamic equilibrium?). Still, it is nice to see how we continue to understand the limits of the principles that we have in physics. At the very least, this paper has a wealth of references to these exotic systems exhibiting such negative specific heat. That alone is enough for me to keep a copy of this paper. :)