What's the criteria for black hole thermodynamically stability? (And dynamical?) It looks like usual criteria (positivity of Hessian; what geometrically means a cancave of entropy) is no useful, becouse entropy is not additive and not extensive for black hole. Then what is the right criteria?
 A: One can see that a black hole in thermodynamic equilibrium with its environment is always unstable by looking at the Hawking temperature of a black hole, given by 
$$
T = \frac{\hbar c^3}{8\pi k_B M G}
$$
The temperature being inversely proportional to the mass means that big black holes are cold, a black hole with the mass of the sun has for example a temperature of about only $4\cdot 10^7 K$. This is much colder than the empty space with a microwave background radiation of about $2,7 K$. So heat (=energy= mass) will flow from empty space which makes it even colder according to the equation for the Hawking radiation.
If the black hole is small and initially warmer than the empty space, heat flows out of the black hole which increases the temperature further. 
This counterintuitive behavior means that the black hole has a negative heat capacity which means that a black hole in thermodynamic equilibrium is unstable.
Lenny Susskind explains these things in the second part of this lecture, my explanations follow what he said there.
