Does any substance melt or evaporate when cooled at constant pressure? Is there any substance with segments of its phase change diagram lines going in a negative direction?
To explain:
Generally, as phase change diagrams go, with heat increasing, and pressure constant, substances tend to evaporate, sublimate or melt; cooling produces the opposite, fusion (freezing), deposit or condensation into liquid from vapor.
But the diagrams are rarely straightforward - water, for example, has a multitude of forms of ice, where not all forms are obtainable through transitions "from any direction". There are many obscure factors that define when each transition can occur.
Is there any substance, that - without transforming into another substance (say, polymerization) - possesses an area of the phase change diagram (possibly way off "room conditions") where the transition goes in the opposite direction - heating leads to a - not necessarily more dense - but 'more solid' state? Something like thermally hardened glue, but without a chemical transition?
 A: This question has already been answered pretty thoroughly, but I feel like I should add that all of this is related to something called the Clausius-Clapeyron equation,
$$
\frac{\textrm{d} P}{\textrm{d} T} = \frac{\Delta s}{\Delta v}
$$
This relates the slope of the phase boundary in a $P-T$ phase diagram to specific entropy change $\Delta s$ and the specific volume change $\Delta v$ of the phase transition. Now, if we talk about going from a solid phase to a liquid phase, $\Delta s$ is always positive, because solids have crystalline order, which is lost when going to the liquid phase. Since the specific volume in this case is usually also positive, we expect the slope of the phase boundary to be positive. In the cases when it is not, we have the very unusual scenario of a substance being more "compact" in liquid phase than in solid phase. That this is true of water is well-known: try putting water in a tight container into your freezer and see how it goes. From lemon's linked image, we see it is also true of helium 3. 
A: Just to follow up John Rennie's comment (because I didn't quite believe it)

Looks like ice between 0 and -20C will melt at some (rather extreme) range of pressures but only at increasing pressure not with decreasing temperature.
A: Yes, here's the phase diagram for Helium-3:

Notice that around 3 MPa, an increase in temperature causes a transition from liquid to solid.
