Stability of linear dilaton black holes at the Hagedorn temperature

We study the thermodynamics and the small fluctuations of a linear dilaton black hole, in an S-dual version of the near-horizon limit of type IIB NS5 branes. The thermodynamical analysis shows that the black hole is in a Hagedorn phase, with marginal stability. The dynamical analysis confirms that the speed of sound of the dual theory vanishes and that there are no evident instabilities at the level of supergravity. We clarify the physical meaning of some singularities of the retarded correlator that were thought to lead to instabilities.