There are a couple of possibilities here. Do you know what year this AVR was produced? Vintage of the AVR may be useful to know.
1) Audio output is delayed upon startup by design. This is to avoid annoying pops, thumps, etc. Typically, you turn on the AVR, then there is several seconds of delay as the caps charge and the AVR gets the initial in-rush current, and internal checks are performed. If all the internal tests pass, then you hear the relays click and you get audio to the speakers. I believe there is some time delay before the tests are performed. If that time delay is based on a charging capacitor, then a cap going bad could be the problem (I don't know if the timer is based on this or not, but it is one way to do this).
2) HDMI requires a handshake between all the gear to satisfy the security screening to be sure you are not trying to bootleg media. Depending on your particular gear, some may take longer than others to perform the handshake.
Do you have the audio problem if you change to a different input, then back to the HDMI input? If you do, then item #2 is likely, but I would expect it to take the same time every time you change. If the problem is only at power up, then I lean more towards item #1.
@TLS Guy what you think?