"About a year ago, we ran a lubricant too long in a high-duty gearbox, and it oxidized and threw sludge. Since then, we've been watching the oil more closely with oil analysis. My problem is that we are now seeing acid numbers increase and oil darken after only one month of service. The lubricant used to last a full year. We keep changing the oil, but the problem just repeats. Why does our gear oil have such a short life?"
It sounds like the gearbox was not thoroughly flushed after the oil oxidized the first time. Sometimes a simple drain will leave more than 15 percent of the old oil behind, occluding to machine surfaces and becoming trapped within the casing. This also leaves behind a host of reactive chemicals (pro-oxidants) that rapidly deplete antioxidant additives, leaving the base oil unprotected.
You refer to the gearbox as high-duty, which probably means high temperature and high wear metal production. The temperature and wear particles also accelerate the rate of oxidation, especially when sludge and other pro-oxidants are in the mix. Consider doing a thorough flushing of the gearbox.