  At zero temperature, spring networks with connectivity below Maxwell's
isostatic threshold undergo a mechanical phase transition from a floppy state
at small strains to a rigid state for applied shear strain above a critical
strain threshold. Disordered networks in the floppy mechanical regime can be
stabilized by entropic effects at finite temperature. We develop a scaling
theory based on a real-space renormalization approach for this mechanical phase
transition at finite temperature, yielding relationships between various
scaling exponents. Using Monte Carlo simulations, we verify these scaling
relations and identify anomalous entropic elasticity with sub-linear
$T$-dependence in the linear elastic regime. While our results are consistent
prior studies of phase behavior near the isostatic point, the present work also
makes predictions relevant to the broad class of disordered thermal
semiflexible polymer networks for which the connectivity generally lies far
below the isostatic threshold.
