  We estimate the accuracy in the measurement of the tidal Love number of a
supermassive compact object through the detection of an extreme mass ratio
inspiral~(EMRI) by the future LISA mission. A nonzero Love number would be a
smoking gun for departures from the classical black hole prediction of General
Relativity. We find that an EMRI detection by LISA could set constraints on the
tidal Love number of a spinning central object with dimensionless spin $\hat
a=0.9$ ($\hat a=0.99$) which are approximately four (six) orders of magnitude
more stringent than what achievable with current ground-based detectors for
stellar-mass binaries. Our approach is based on the stationary phase
approximation to obtain approximate but accurate semi-analytical EMRI waveforms
in the frequency-domain, which greatly speeds up high-precision
Fisher-information matrix computations. This approach can be easily extended to
several other tests of gravity with EMRIs and to efficiently account for
multiple deviations in the waveform at the same time.
