Reissner-Nordstrom is thermodynamically unstable, and so are the Schwarzschild and Kerr solutions. The thermodynamic instability is easy to see and explain: smaller black holes are hotter than larger ones. So as a black hole radiates via the Hawking process, the black hole losses mass, shrinks, and becomes hotter. If you tried to put these black holes into thermal equilibrium with a heat bath, then any fluctuation of temperature would result in the black hole either evaporating to zero size or expanding to infinite size (depending on whether the initial fluctuation made the black hole a bit hotter, or a bit cooler than the heat bath, respectively).
This instability is not a problem the astrophysical relevance of black holes--it simply means that a truly time independent black hole isn't sensible--all black holes are either radiating or absorbing radiation.
The instability is strongly connected to the boundary conditions, in this case asymptotic flatness. If the black holes were instead in AdS space, then thermal equilibrium is possible. In fact there is a whole rich story of the thermodynamics of black holes in AdS, which has a strong connection with the AdS/CFT correspondence. The most famous phase transition in this context (which actually predates the AdS/CFT correspondence) is the Hawking-Page transition http://projecteuclid.org/euclid.cmp/1103922135.