We study the elastic properties of thermal networks of Hookean springs. In the purely mechanical limit, such systems are known to have vanishing rigidity when their connectivity falls below a critical, isostatic value. In this work we show that thermal networks exhibit a non-zero shear modulus G well below the isostatic point, and that this modulus exhibits an anomalous, sublinear dependence on temperature T. At the isostatic point, G increases as the square-root of T, while we find G ∝ T^α

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... w the isostatic point, where α≃ 0.8. We show that this anomalous T dependence is entropic in origin.