  Van der Waals moir\'e materials have emerged as a highly controllable
platform to study the electronic correlation phenomena. In particular, robust
correlated insulating states have recently been discovered at both integer and
fractional filling factors of the semiconductor moir\'e systems. Here, we
reveal the thermodynamic properties of these states by measuring the gate
capacitance of MoSe2/WS2 moir\'e superlattices. We observe a series of
incompressible states for filling factor 0 - 8 and anomalously large
capacitance (nearly 60% above the device's geometrical capacitance) in the
intervening compressible regions. The anomalously large capacitance is most
pronounced at small filling factor, below the melting temperature of the
charge-ordered states, and for small sample-gate separation. It is a
manifestation of the device-geometry-dependent Coulomb interaction between
electrons and phase mixing of the charge-ordered states. We have further
extracted the thermodynamic gap of the correlated insulating states and the
entropy of the capacitive device. The results not only establish capacitance as
a powerful probe of the correlated states in semiconductor moir\'e systems, but
also demonstrate control of the extended Coulomb interaction in these materials
via sample-gate coupling.
