  The leading $(\alpha')^3$-correction to the gravitational low-energy
effective action of closed (type II) superstring theory in four-spacetime
dimensions defines the Einstein-Grisaru-Zanon gravity action that is applied
for a calculation of the leading corrections to the Schwarzschild solution and
the Hubble function in the Friedmann-Lemaitre-Robertson-Walker universe, in the
first order with respect to the effective string-generated coupling. The
solutions found are compared to the corresponding solutions in the
Einstein-Bel-Robinson gravity that also modifies the Einstein gravity by the
terms quartic in the spacetime curvature. We consider the black hole shadows in
the Einstein-Grisaru-Zanon gravity theory and derive the upper bound on the
string coupling parameter from the Hawking temperature of a black hole.
