  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 $G \propto T^{\alpha}$ below the isostatic
point, where ${\alpha} \simeq 0.8$. We show that this anomalous $T$ dependence
is entropic in origin.
