  We consider a random Hamiltonian $H:\Sigma\to\mathbb R$ defined on a compact
space $\Sigma$ that admits a transitive action by a compact group $\mathcal G$.
When the law of $H$ is $\mathcal G$-invariant, we show its expected free energy
relative to the unique $\mathcal G$-invariant probability measure on $\Sigma$
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.
