Classical and thermodynamic stability of black branes

It is argued that many nonextremal 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.