  The Laser Interferometer Space Antenna (LISA) will open the mHz frequency
window of the gravitational wave (GW) landscape. Among all the new GW sources
expected to emit in this frequency band, extreme mass-ratio inspirals (EMRIs)
constitute a unique laboratory for astrophysics and fundamental physics. Here
we show that EMRIs can also be used to extract relevant cosmological
information, complementary to both electromagnetic (EM) and other GW
observations. By using the loudest EMRIs (SNR$>$100) detected by LISA as dark
standard sirens, statistically matching their sky localisation region with mock
galaxy catalogs, we find that constraints on $H_0$ can reach $\sim$1.1%
($\sim$3.6%) accuracy, at the 90% credible level, in our best (worst) case
scenario. By considering a dynamical dark energy (DE) cosmological model, with
$\Lambda$CDM parameters fixed by other observations, we further show that in
our best (worst) case scenario $\sim$5.9% ($\sim$12.3%) relative uncertainties
at the 90% credible level can be obtained on $w_0$, the DE equation of state
parameter. Besides being relevant in their own right, EMRI measurements will be
affected by different systematics compared to both EM and ground-based GW
observations. Cross validation with complementary cosmological measurements
will therefore be of paramount importance, especially if convincing evidence of
physics beyond $\Lambda$CDM emerges from future observations.
