  A dominant rational self-map on a projective variety is called
$p$-cohomologically hyperbolic if the $p$-th dynamical degree is strictly
larger than other dynamical degrees. For such a map defined over
$\overline{\mathbb{Q}}$, we study lower bounds of the arithmetic degrees,
existence of points with Zariski dense orbit, and finiteness of preperiodic
points. In particular, we prove that, if $f$ is an $1$-cohomologically
hyperbolic map on a smooth projective variety, then (1) the arithmetic degree
of a $\overline{\mathbb{Q}}$-point with generic $f$-orbit is equal to the first
dynamical degree of $f$; and (2) there are $\overline{\mathbb{Q}}$-points with
generic $f$-orbit. Applying our theorem to the recently constructed rational
map with transcendental dynamical degree, we confirm that the arithmetic degree
can be transcendental.
