Van der Waals moiré materials have emerged as a highly controllable platform to study electronic correlation phenomena1-17. Robust correlated insulating states have recently been discovered at both integer and fractional filling factors of semiconductor moiré systems10-17. In this study we explored the thermodynamic properties of these states by measuring the gate capacitance of MoSe2/WS2 moiré superlattices. We observed a series of incompressible states for filling factors 0-8 and anomalously large capacitance in the intervening compressible regions. The anomalously large capacitance, which was nearly 60% above the device's geometrical capacitance, was most pronounced at small filling factors, 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. Based on these results, we were able to extract the thermodynamic gap of the correlated insulating states and the device's electronic entropy and specific heat capacity. Our findings establish capacitance as a powerful probe of the correlated states in semiconductor moiré systems and demonstrate control of these states via sample-gate coupling.
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