  We construct endomotives associated to toric varieties, in terms of the
decomposition of a toric variety into torus orbits and the action of a
semigroup of toric morphisms. We show that the endomotives can be endowed with
time evolutions and we discuss the resulting quantum statistical mechanical
systems. We show that, in particular, one can construct a time evolution
related to the logarithmic height function. We discuss relations to
F1-geometry.
