  We propose a method for the measurement of adiabatic phases of periodically
driven quantum systems coupled to an open cavity that enables dispersive
readout. It turns out that the cavity transmission exhibits peaks at
frequencies determined by a resonance condition that involves the dynamical and
the geometric phase. Since these phases scale differently with the driving
frequency, one can determine them by fitting the peak positions to the
theoretically expected behavior. For the derivation of the resonance condition
and for a numerical study, we develop a Floquet theory for the dispersive
readout of ac-driven quantum systems. The feasibility is demonstrated for two
test cases that generalize Landau-Zener-St\"uckelberg-Majorana interference to
two-parameter driving.
