BMW does an extraordinary amount of cold weather testing so that the M5 performs the best it can in severely cold environments.
The loud engine tone for about under one minute is normal upon cold starts.
The OP's issue seems to be uncharacteristic and I would bring it in for a diagnosis and read-out.
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2013 BMW INDIVIDUAL ///M5 {Frozen Black/Amaro Brown/Amaro Brown Alcantara/White Contrast Stitching/Piano Black Trim}
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