We continue the study of the arithmetic geometry of toric varieties started by J. Burgos Gil, P. Philippon, and M. Sombra in 2011. In this text, we study the positivity properties of metrized
R
\mathbb {R}
-divisors in the toric setting. For a toric metrized
R
\mathbb {R}
-divisor, we give formulae for its arithmetic volume and its
χ
\chi
-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a higher-dimensional analogue of Dirichlet’s unit theorem for toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized
R
\mathbb {R}
-divisor, and we prove a toric arithmetic Fujita approximation theorem.