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