  Possibilities for the measurement of the longitudinal structure function in
diffraction $F_\mathrm{L}^\mathrm{D}$ at the future US Electron Ion Collider
are investigated. The sensitivity to $F_\mathrm{L}^\mathrm{D}$ arises from the
variation of the reduced diffractive cross section with centre-of-mass energy.
Simulations are performed with various sets of beam energy combinations and for
different assumptions on the precision of the diffractive cross section
measurements. Scenarios compatible with current EIC performance expectations
lead to an unprecedented precision on $F_\mathrm{L}^\mathrm{D}$ at the 5-10 %
level in the best measured regions. While scenarios with data at a larger
number of centre-of-mass energies allow the extraction of
$F_\mathrm{L}^\mathrm{D}$ in the widest kinematic domain and with the smallest
uncertainties, even the more conservative assumptions lead to precise
measurements. The ratio $R^\mathrm{D}$ of photoabsorption cross sections for
longitudinally to transversely polarised photons can also be obtained with high
precision using a separate extraction method.
