  Two new fast single radio bursts FRB 180924 and FRB 190523 well localized to
massive galaxies have opened a new window to probe and characterize how cosmic
baryons are allocated between galaxies, their surroundings and intergalactic
medium. We are motivated by testing Einstein's weak equivalence principle with
these two cosmic transients which have accurate redshifts. Using photons with
different energies emitted by FRB 180924, we obtain, so far, the most stringent
bound $\Delta\gamma<2.16\times10^{-10}$ for non-repeating FRBs with accurate
redshifts when only considering the gravitational potential of the Milk Way. If
using the gravitational potential of the Laniakea supercluster instead of the
Milk Way one, we also obtain the strictest bound
$\Delta\gamma<1.06\times10^{-14}$ to date. In light of rapid progress of FRB
cosmology, towards the next two decades, we give an universal limitation
$\Delta\gamma<8.24\times10^{-22}$ from photons with different energies emitted
by single FRBs with accurate redshifts. Moreover, we analyze detailedly the
effects of various astrophysical parameters on the precision of weak
equivalence principle. We also estimate the abilities of single FRBs with known
redshifts to test the validity of swampland criterion, and to distinguish which
value of $H_0$ is preferred.
