In conventional metals, quantum oscillations arise from Landau quantization of Fermi-surface cyclotron orbits, whose dynamics are governed by the Fermi velocity and cyclotron effective mass within Lifshitz-Kosevich (LK) theory. A perfectly flat band, by contrast, has vanishing group velocity, which would naively imply an infinite cyclotron mass and complete thermal suppression of quantum oscillations. Yet topological flat bands can support anomalous Landau levels (LLs) whose finite-field spacing is generated by quantum geometry rather than band curvature, allowing quantum oscillations to persist. This work addresses how such anomalous flat-band LLs behave within the LK framework and whether their thermal damping can reveal quantum geometric information. Using a minimal model with exactly flat topological bands, we derive an LK theory for these anomalous LLs and analyze fixed-density magnetization oscillations. The resulting oscillations exhibit a finite LK effective mass that is substantially larger than the normal-band value and possesses a strong magnetic-field dependence. In the weak-field limit, this anomalous mass reflects the quantum geometric origin of the LL spacing and scales inversely with both the magnetic field and the trace of the quantum metric. Thus, thermal damping of flat-band quantum oscillations directly measures the quantum metric, establishing quantum oscillations as a probe to flat-band quantum geometry.