Guessing the temperature dependence of a decay rate $\Gamma(A\to B+B)$

For a two-body decay of the form $$A\to B+B$$ if the interaction strength controlling the decay is $$\lambda$$, the Feynman amplitude $$\mathcal{M}$$ will contain a factor of $$\lambda$$ from the vertex factor $$(-i\lambda)$$. Therefore, the zero-temperature decay rate $$\Gamma_{\rm T=0}$$ will be given by $$\Gamma_{\rm T=0}\sim |\mathcal{M}|^2\sim \lambda^2.$$ If the same decay occurs at a temperature $$T\neq0$$, what can be said about the $$T-$$dependence of the decay rate (without a detailed calculation of thermal field theory) at least for temperatures $$T\gg M_A$$? I am looking for some plausibility arguments based on dimensional analysis or otherwise.

For a Yukawa interaction of the type $$\mathscr{H}_{int}=-\lambda \bar{\psi}_B\psi_B\phi_A,\tag{1}$$ where $$\phi$$ is a scalar field of mass $$M_A$$ and $$\psi$$ is a fermionic field of mass $$M_B$$ (with $$M_A>2M_B$$), is responsible for a decay of the form $$A\to B+\bar{B}.\tag{2}$$ The exact zero-temperature decay rate at the tree-level is given by $$\Gamma_{\rm T=0}=\frac{\lambda^2M_A}{8\pi^2}\Big(1-\frac{4M_B^2}{M_A^2}\Big)^2.\tag{3}$$ In the limit where $$M_B$$ mass can be neglected, the relevant parameters of the problem are $$\lambda$$ and $$M_A$$. Hence, from dimensional analysis, $$\Gamma_{\rm T=0}=\lambda^2M_A f(M_B/M_A)\tag{4}$$ where $$f$$ is a function of dimensionless quantity $$M_B/M_A$$. Now at finite temperature $$T$$, where both $$M_A,M_B$$ can be neglected, dimensional analysis tells that $$\Gamma_{\rm T\neq 0}\sim \lambda^2T.\tag{5}$$ If someone wants to refine $$(5)$$ when $$M_A,M_B$$ are not neglected that would be even more interesting.