  We consider dynamical systems on compact manifolds, which are local
diffeomorphisms outside an exceptional set (a compact submanifold). We are
interested in analyzing the relation between the integrability (with respect to
Lebesgue measure) of the first hyperbolic time map and the existence of
positive frequency of hyperbolic times. We show that some (strong)
integrability of the first hyperbolic time map implies the existence of a
positive Lebesgue measure subset of points with positive frequency of
hyperbolic times. We also present an example of a map with positive frequency
of hyperbolic times at Lebesgue almost every point but whose first hyperbolic
time map is not integrable with respect to the Lebesgue measure.
