r$_{+}$ < r$_{1ext}$ or r$_{+}$ > r$_{2ext}$ for hyperbolic black branes. We found that the existence of r$_{ext}$ for k = 0, and r$_{1ext}$ for k = − 1 depends on the value of α . For the case of k = 0 with large values of α ( α > α$_{ext}$ ), one can have black brane with any size. This feature shows that dilaton change the thermodynamics of black branes drastically. In this case, the temperature goes to infinity as r$_{+}$ goes to zero for n > 4, while it approaches to a constant for n = 4. For the case of k = − 1, one may have small black branes provided α$_{1ext}$ < α < α$_{2ext}$ , while large black branes exist for any value of α . Again, this is a drastic change in the properties of the solutions because of the dilaton field. We analyzed the thermal stability of the solutions in the canonical ensemble by finding a Smarr-type formula and considering ( ∂ $^{2}$M/∂S$^{2}$)$_{Q}$ for the charged topological dilaton black brane solutions in ( n +1) dimensions. We showed that for the case of k = 0, small black branes are unstable for α$_{ext}$ < α < α$_{crit}$ , while they are stable for α > α$_{crit}$ . Also, in this case large black branes are stable for arbitrary α . For the case of k = − 1, small black branes are unstable, while large ones are stable. That is dilaton with the potential given in Eq. (2), changes the stability of the black branes drastically. 

## Acknowledgments 

We thank the referees for constructive comments. This work has been supported by Research Institute for Astronomy and Astrophysics of Maragha, Iran. 

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