From knowledge of other cars, but no problems yet in this area on my 2001 3.0i ...

the secondary air pump is designed to come on when the engine is stone cold (e.g., below about 60*F, so even a warm day can make it not needed), to pump extra air into the exhaust to complete combustion and bring the cat converters up to temp as quickly as possible. They run for a set time, like 60 or 90 seconds, then stop until the next stone cold start.

For cars that might not have these, sometimes you might detect a faint gas smell coming from the exhaust immediately following a cold start. That's what the SAP is intended to mitigate.

So it's an emissions optimization thing that only even tries to matter in the first minute or so of a cold start.

If it were me, I'd surely track it down to confirm where the unknown noise was coming from, but if I found it to be the secondary air pump / air pump / smog pump ... just between you and me ... I might not make it a top priority to fix it.

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2001 X5 3.0i, 203k miles, AT, owned since 2014