  We study the elasticity of thermalized spring networks under an applied bulk
strain. The networks considered are sub-isostatic random-bond networks that, in
the athermal limit, are known to have vanishing bulk and linear shear moduli at
zero bulk strain. Above a bulk strain threshold, however, these networks become
rigid, although surprisingly the shear modulus remains zero until a second,
higher, strain threshold. We find that thermal fluctuations stabilize all
networks below the rigidity transition, resulting in systems with both finite
bulk and shear moduli. Our results show a $T^{0.66}$ temperature dependence of
the moduli in the region below the bulk strain threshold, resulting in networks
with anomalously high rigidity as compared to ordinary entropic elasticity.
Furthermore we find a second regime of anomalous temperature scaling for the
shear modulus at its zero-temperature rigidity point, where it scales as
$T^{0.5}$, behavior that is absent for the bulk modulus since its athermal
rigidity transition is discontinuous.
