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.This work was supported by the Fundação para a Ciência e Tecnologia (Portugal), throught the grant SFRH/BD/22211/2005, and, in its final stages, by the Cambridge Philosophical Society
