Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$

Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that $[D']=\pi_*(a[D])$ is an invariant Cartier divisor on $X$ (associated to some ray in the fan of $X$).

Question: For any integer $m$, is it true that $H^i(\mathcal{X}, \mathcal{O}(mD)) \cong H^i(X, \mathcal{O}(qD'))$, where $q$ is the ceiling of $m/a$ (i.e. the smallest integer number such that $m/a\leq q$)?

P.S.: I do not know so much about (toric) stacks, hence I do not know if my question is well-posed. Please, suggest me any kind of corrections.