  We consider the possible observation of Fast Radio Bursts (FRBs) with planned
future radio telescopes, and investigate how well the dispersions and redshifts
of these signals might constrain cosmological parameters. We construct mock
catalogues of FRB dispersion measure (DM) data and employ Markov Chain Monte
Carlo (MCMC) analysis, with which we forecast and compare with existing
constraints in the flat $\Lambda$CDM model, as well as some popular extensions
that include dark energy equation of state and curvature parameters. We find
that the scatter in DM observations caused by inhomogeneities in the
intergalactic medium (IGM) poses a big challenge to the utility of FRBs as a
cosmic probe. Only in the most optimistic case, with a high number of events
and low IGM variance, do FRBs aid in improving current constraints. In
particular, when FRBs are combined with CMB+BAO+SNe+$H_0$ data, we find the
biggest improvement comes in the $\Omega_{\mathrm b}h^2$ constraint. Also, we
find that the dark energy equation of state is poorly constrained, while the
constraint on the curvature parameter $\Omega_k$, shows some improvement when
combined with current constraints. When FRBs are combined with future BAO data
from 21cm Intensity Mapping (IM), we find little improvement over the
constraints from BAOs alone. However, the inclusion of FRBs introduces an
additional parameter constraint, $\Omega_{\mathrm b}h^2$, which turns out to be
comparable to existing constraints. This suggest that FRBs provide valuable
information about the cosmological baryon density in the intermediate redshift
Universe, independent of high redshift CMB data.
