  We cast a nonzero-temperature analysis of the jamming transition into the
framework of a scaling ansatz. We show that four distinct regimes for scaling
exponents of thermodynamic derivatives of the free energy such as pressure,
bulk and shear moduli, can be consolidated by introducing a universal scaling
function with two branches. Both the original analysis and the scaling theory
assume that the system always resides in a single basis in the energy
landscape. The two branches are separated by a line $T^*(\Delta \phi)$ in the
$T-\Delta \phi$ plane, where $\Delta \phi=\phi-\phi_c^\Lambda$ is the deviation
of the packing fraction from its critical, jamming value, $\phi_c^\Lambda$, for
that basin. The branch for $T<T^*(\Delta \phi)$ reduces at $T=0$ to an earlier
scaling ansatz that is restricted to $T=0$, $\Delta \phi \ge 0$, while the
branch for $T>T^*(\Delta \phi)$ reproduces exponents observed for thermal hard
spheres. In contrast to the usual scenario for critical phenomena, the two
branches are characterized by different exponents. We suggest that this unusual
feature can be resolved by the existence of a dangerous irrelevant variable
$u$, which can appear to modify exponents if the leading $u=0$ term is
sufficiently small in the regime described by one of the two branches of the
scaling function.
