We consider a random Hamiltonian H:Σ→R defined on a compact
space Σ that admits a transitive action by a compact group G.
When the law of H is G-invariant, we show its expected free energy
relative to the unique G-invariant probability measure on Σ
obeys a subadditivity property in the law of H itself. The bound is often
tight for weak disorder and relates free energies at different temperatures
when H is a Gaussian process. Many examples are discussed including branching
random walk, several spin glasses, random constraint satisfaction problems, and
the random field Ising model. We also provide a generalization to quantum
Hamiltonians with applications to the quantum SK and SYK models