FIG. 9: 10 − $^{2}$( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ (solid line) and T (dashed line) versus α for b = 0 . 2, n = 4, l = 1 and r$_{+}$ = 0 . 1. FIG. 10: 10 $^{2}$( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ (solid line) and T (dashed line) versus α for b = 0 . 2, n = 4, l = 1 and r$_{+}$ = 5. on the ensemble that is used. In the canonical ensemble, the charge is a fixed parameter and therefore the positivity of the ( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ is sufficient to ensure local stability. In Figs. 7 - 10 we show the behavior of the ( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ as a function of the coupling constant parameter α for different values of the size of black brane r$_{+}$ . In order to investigate the stability of black branes, we plot both ( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ and T in one single figure for various values of r$_{+}$ or α . Of course, one should note that in these figures, only the positive values of temperature associated to the event horizon of the black branes, and negative values of temperature belong to inner horizon which we are not interested in. We discuss these figures for k = 0 and k = − 1, separately. 15 k = –1 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 alpha k = 0 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 alpha k = –1 –1.4 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.5 1 1.5 2 2.5 alpha k = 0 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 alpha