Let $\mathbb{F}^m=(M,F)$ be a Finsler manifold and $G$ be the Sasaki-Finsler metric on the slit tangent bundle $TM^0=TM\setminus\{0\}$ of $M$. We express the scalar curvature $\widetilde\rho$ of the Riemannian manifold $(TM^0,G)$ in terms of some geometrical objects of the Finsler manifold $\mathbb{F}^m$. Then, we find necessary and sufficient conditions for $\widetilde\rho$ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of $TM^0$. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose $\widetilde\rho$ satisfies the above condition.