Classical and Thermodynamic Stability of Black Branes

It is argued that many non-extremal black branes exhibit a classical Gregory-Laflamme instability if, and only if, they are locally thermodynamically unstable. For some black branes, the Gregory-Laflamme instability must therefore disappear near extremality. For the black p-branes of the type II supergravity theories, the Gregory-Laflamme instability disappears near extremality for p = 1, 2, 4 but persists all the way down to extremality for p = 5, 6 (the black D3-brane is not covered by the analysis of this paper). This implies that the instability also vanishes for the near-extremal black M2 and M5-brane solutions.