  We study, for the first time, differential distributions for two $\rho^0$
meson production in exclusive ultraperipheral, ultrarelativistic heavy ion
collisions via a double-scattering mechanism. The calculations are done in the
impact parameter space. The cross section for $\gamma A \to \rho^0 A$ is
parametrized based on an existing model. Smearing of $\rho^0$ masses is taken
into account. The results of calculations for single and double-$\rho^0$
production are compared to experimental data at the RHIC and LHC energies. The
mechanism considered gives a significant contribution to the $AA \to AA
\pi^+\pi^-\pi^+\pi^-$ reaction. Some observables related to charged pions are
presented too. We compare results of our calculations with the STAR
collaboration results for four charged pion production.
