The band gap of Diamond, Silicon and Germanium is 5.5, 1.12, 0.66 eV while the average energy to produce an electron-hole pair for the same semiconductor is namely 13, 3.6 and 2.9 eV. This ratios are ranging from $\sim4,4$ to $\sim2$. Is there a simple rule connecting all indirect semiconductors band gaps to their electron-hole pair generation threshold? I cannot find any reference paper discussing this specific relation...
I finally found a paper addressing this correspondance between band gap and electron hole pair energies. I hope this will help some other.
As you can read on the paper there is such a simple law to link the two. You may refer to the image:
The deal here is to use momentum conservation. Read the paper for further infos (don't want to break some copyright).