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I heard that this statement is correct. However, it seems odd to me. The number of possible microstates is still the same, so isn't the entropy constant?

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The number of microstates that may represent a macroscopic configuration behaves like $$ N\sim\exp(S/k)$$ so it is not constant if the entropy increases. What you're probably missing is that the microstates are defined not only by positions but by velocities (or momenta), too.

Take $N$ molecules of a gas. They may be spread in the volume $V$ of the ordinary space but it's also important that they have some average velocities $v_{\rm rms}$ so all the molecules are localized not only in the volume $V$ of the position space but also in the volume $V_p\sim (mv_{\rm rms})^3$ of the momentum space. Because the higher temperature increases the average velocity or momentum, it increases the number of states because it increases the volume in the momentum space in which the particles are "confined" much like in the position space.

Quantum mechanic says that there is one microstate per volume $(2\pi \hbar)^N$ where $N$ is the number of independent coordinates (and the number of independent momenta). But this $(2\pi\hbar)^N$ is a volume measured in the whole phase space that has all the position and momentum components as the coordinates, not just a volume in the ordinary position space.

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