Thermodynamically speaking

In the context of reversible unfolding, protein stability is defined as the difference between the free energies of the unfolded and folded states, ΔG. That is, as long as we can safely talk about a ‘two state’ unfolding process. 

So the greater this difference, the more stable the protein. 

Although it is easy to derive the formula that gives ΔG with respect to temperature (see the curve in the figure below) it is much harder, and not always possible, to experimentally determine the two parameters of the formula that differ for different proteins and determine the exact shape of the curve (for an enlightening discussion on thermodynamic stability see the relevant section of this review or the original work of Nojima et al.).

Typical stability curve of a protein (G. Feller, J. Phys.: Condens. Matter, 2010)

A couple of months ago, there came to light a very interesting work, by C.C. Liu and V.J. LiCata, on the detailed thermodynamic study of the thermal stability of two highly structurally homologous proteins. The thermophilic Taq polymerase and its homologous mesophilic Pol I polymerase from E.Coli. The authors, by decomposing the ΔG curve into its two competing enthalpic and entropic components, show that the increased stability of Taq polymerase is entropic in nature. But, lo and behold, this pair is not the only such case. In fact, the authors use the same analysis on another 17 available pairs of homologous proteins, which are pretty much all the existing published data there are for which this analysis is applicable. In almost all the cases, for the thermophilic homologue the stabilizing enthalpic contribution has to compensate a smaller entropic penalty than for the mesophilic one. Why “penalty”? Well, in all temperatures above the maximal stability temperature, the entropic contribution is unfavorable for the folded state (look at the signs of ΔH and TΔS in the figure above and remember that ΔG=ΔΗ-ΤΔS). 
Now, the entropy of protein folding in general has two major opposite contributions: the favorable hydrophobic effect and the unfavorable loss of conformational entropy that comes with the protein collapse. Thus when thermophiles have to compensate a smaller entropic penalty, they either have a rather compact or structured denatured state that already isolates the hydrophobic groups from the water or … a more flexible folded state. Or both.