In popular Theravada, one encounters the term apāyamukha reasonably frequently as a blanket term for kinds of conduct that lead to lower rebirths. In this reading, apāya means “lower realm” and mukha means “cause”.
The term is occasionally encountered in the EBTs, but it seems that it never has this sense. Rather, in AN 4.178, AN 8.54, and AN 8.55 it is used in the sense of “drain”, that is, an “opening” (mukha) for “leaking out” apāya. The basic application of it is to the things that are a “drain” on wealth, such as gambling, etc. In this sense it also occurs in DN 31. In AN 4.178 and DN 3 the metaphor is applied somewhat differently, but the basic idea is the same. In none of these contexts does apāya mean “lower realm”, nor does the phrase have anything to do with the causes of rebirth.
It seems that the popular sense must have emerged later on. Anyway, it’s useful to know such things when reading the suttas. Someone should really compile a list of terms that have different meanings in the suttas and later. <sarcasm>I’m sure it would not be controversial at all!</sarcasm>
Apāya, as well as duggati and other words, does indeed mean lower realm; this is the most common meaning in the EBTs. It just doesn’t mean it in this context.
Return of the mukha!
I’m sure you remember our debate about ‘parimukha’ bhante. At least my conclusion from that discussion was that indeed ‘-mukha’ in a compound meaning ‘opening’ and not as usually understood ‘mouth’ or even ‘front’.
Be it as it may, here it obviously fits as ‘opening’ very well.
Regarding apāya, apa can also have the meaning of ‘down’. Then apāyamukha would mean ‘opening-to-go-down’ or so. It would in that sense be a nice antonym to utpad, as in opapātika, who would be one-who-is-flying-up!
Well, its derivation is very general, from apa- “away from, off of” (which is often conflated with ava- “down”) together with i, go. So it just means “out-goer”, “down-goer”, and could in principle have any number of applications. Having said which, in the EBTs it is, I think, restricted to these two senses.