Thermal density matrix breaks down the Page curve

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In this paper, we study entanglement islands and the Page curve in the eternal four-dimensional Schwarzschild black hole surrounded by finite temperature conformal matter. By finite temperature conformal matter, we mean the matter described by the thermal density matrix, rather than the usually considered matter in the vacuum state. We take the matter and the black hole at different temperatures and calculate the entanglement entropy for such a setup using the s-wave approximation. As a result, we obtain that at late times the island prescription leads to the exponential growth of the entanglement entropy of conformal matter in this thermal vacuum.