k = 0: As we discussed in the last section, small black brane with a radius r$_{+}$ exist when α > α$_{ext}$ . Figures 7 and 8 show that these small black branes are stable provided α > α$_{crit}$ . On the other side, large black branes are stable as one may see in Figs. 9 and 10. k = − 1: Figures 7 and 8 show that small black branes exist only for medium values of α ( α$_{1ext}$ < α < α$_{2ext}$ ), but they are unstable. On the other side, large black branes are stable as one may see in Figs. 9 and 10. In order to confirm the stability analysis of the black branes, one can also use the gener- alized free energy [38] which applies to any value of r$_{+}$ with a fixed temperature T . This off-shell free energy reduces to the on-shell free energy at T = T$_{+}$ : F = M − T$_{+}$S$_{+}$, F $^{off}$( b, α, r$_{+}$, T ) = M ( b, α, r$_{+}$ ) − T S$_{+}$ ( b, α, r$_{+}$ ) , 16 which is the Legendre transform of M with respect to S . As an example of using the stability analysis by off-shell free energy, we plot F off versus r$_{+}$ for k = 0, α = 1 . 2 and α = 4. These are plotted in Figs. 11 and 12, which show that the black brane is unstable for α = 1 . 2, while it is stable for α = 4. These figures confirm the stability analysis of Fig. 8. One may also confirm other stability analysis given in this section. FIG. 11: F off versus r$_{+}$ for b = 0 . 2, n = 4, l = 1, k = 0, α = 1 . 2 and T = 0 . 42. 0 0.0005 0.001 0.0015 0.002 0.0025 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 r+