It seems to me that for any complete toric variety $P_\Sigma$, the anticanonical divisor is big. Moreover, the argument also shows that $P_\Sigma$ is projective. This sounds a bit strange for me, did I make any mistake in the argument below?
We can always shrink the coefficient of the anticanonical divisor $$-K_{P_\Sigma} = \sum_{\rho} D_\rho$$ from $1$ to a smaller positive number $a_\rho$ such that $$\sum_{\rho} a_\rho D_\rho$$ is an ample divisor (think about a small ball around the origin, and shrink each ray $\rho$ so that it lands on the boundary of this ball).
Then because $$-K_{P_\Sigma} = \sum_{\rho} a_\rho D_\rho + D$$ where $D$ is a effective divisor, we know $-K_{P_\Sigma}$ is a big divisor.
This argument also shows that complete toric variety is projective (because it always has an ample divisor).