  Integer and fractional Chern insulators have been extensively explored in
correlated flat band models. Recently, the prediction and experimental
observation of fractional quantum anomalous Hall (FQAH) states with spontaneous
time-reversal-symmetry breaking have garnered attention. While the
thermodynamics of integer quantum anomalous Hall (IQAH) states have been
systematically studied, our theoretical knowledge on thermodynamic properties
of FQAH states has been severely limited. Here, we delve into the general
thermodynamic response and collective excitations of both IQAH and FQAH states
within the paradigmatic flat Chern-band model with remote band considered. Our
key findings include: i) In both $\nu$ = 1 IQAH and $\nu$ = 1/3 FQAH states,
even without spin fluctuations, the charge-neutral collective excitations would
lower the onset temperature of these topological states, to a value
significantly smaller than the charge gap, due to band-mixing and
multi-particle scattering; ii) By employing large-scale thermodynamic
simulations in FQAH states in the presence of strong inter-band mixing between
C = $\pm1$ bands, we find that the lowest collective excitations manifest as
the zero-momentum excitons in the IQAH state, whereas in the FQAH state, they
take the form of magneto-rotons with finite momentum; iii) The unique charge
oscillations in FQAH states are exhibited with distinct experimental
signatures, which we propose to detect in future experiments.
