III. THERMODYNAMICS OF ADS DILATON BLACK BRANE In this section we intend to study thermodynamics of topological dilaton black branes in the background of AdS spaces. First of all we focus on entropy. The entropy of the dilaton black hole typically satisfies the so-called area law of the entropy which states that the entropy of the black hole is one-quarter of the event horizon area [30]. This near universal law applies to almost all kinds of black objects, including dilaton black holes, in Einstein gravity [31]. It is a matter of calculation to show that the entropy of the dilaton black brane per unit volume is where Γ$_{+}$ = Γ( r = r$_{+}$ ). It is notable to mention that in contrast with the higher derivative gravities that may lead to negative entropy [32], the presented entropy is positive definite (0 ≤ Γ$_{+}$ ≤ 1). The Hawking temperature of the dilaton black brane on the outer horizon r$_{+}$ , may be obtained through the use of the definition of surface gravity, where χ = ∂$_{t}$ is the killing vector and a prime stands for the derivative with respect to r . Finding the radius of outer horizon in terms of the parameters of the metric function is not possible analytically, and therefore we obtain the constant c in terms of b , α and r$_{h}$ by solving f ( r$_{h}$ ) = 0, where r$_{h}$ is the radius of inner or outer horizon of the black brane. Substituting c into Eq. (21), one obtains where Γ$_{h}$ = Γ( r = r$_{h}$ ). The equation T$_{h}$ = 0 has one real root for k = 0: while it may have two real roots ( r$_{1ext}$ and r$_{2ext}$ ) for k = − 1. These roots are the radius of the extreme black branes. We are interested in the thermodynamics of event horizon, r$_{+}$ , of the black branes and therefore we consider T$_{+}$ ≥ 0. The negative values of T$_{h}$ associated r$_{ext}$ = b { ( ( n − 1) [2 − γ ( n − 2)] n ) 2 / ( n − 2) − 1 } 1 / 2 , (23) T$_{h}$ = b ( n − 2)Γ 1 − γ ( n − 1) / 2 h 4 π (1 − Γ$_{h}$) 1 / ( n − 2) { [ γ ( n − 1) − 1] l $^{2}$Γ 2 γ ( n 1) h − ( n − 1) [ γ ( n − 2) − 2] ( n 2) l $^{2}$Γ 1 γ ( n 1) h + k (1 − Γ$_{h}$) 2 / ( n − 2) b 2 } , (22) T$_{+}$ = 1 2 π √ − 1 2 ( ∇ $_{µ}$χ$_{ν}$ ) ( ∇ $^{µ}$χ$^{ν}$) = √ r 2 + + b 2 ( N $^{2}$f $^{2}$) ′ 4 πNr ∣ r = r + , (21) S = b n − $^{1}$Γ γ ( n − 1) / 2 + 4 (1 − Γ$_{+}$) ( n − 1) / ( n − $_{2)}$, (20) 8