Varying the quantum-well width in an HgTe/CdTe heterostructure allows to realize normal and inverted semiconducting band structures, making it a prototypical system to study two-dimensional (2D) topological-insulator behavior. We have calculated the zero-temperature thermodynamic density of states $D_mathrm{T}$ for the electron-doped situation in both regimes, treating interactions within the Hartree-Fock approximation. A distinctively different behavior for the density dependence of $D_mathrm{T}$ is revealed in the inverted and normal cases, making it possible to detect the systems topological order through measurement of macroscopic observables such as the quantum capacitance or electronic compressibility. Our results establish the 2D electron system in HgTe quantum wells as unique in terms of its collective electronic properties.
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