Spinning Axe

If Temporary is granted to an attached Spinning Axe, will it die at the start of my Beginning Phase?

Yes. The Spinning Axe will have two instances of the Temporary keyword - one that is active and one that is inactive. The inactive keyword will not trigger, but the active one will. Alas, Spinning Axe will die.

Multiple instances of the Temporary keyword are redundant per rule 816.2., but that rule refers to the fact that multiple instances that could trigger will only trigger once, as clarified by rule 816.2.a. In the case of granted Temporary on an attached Spinning Axe, that rule isn't of any concern - there is only one instance of Temporary that could trigger, so it will.

It can help to visualize the attached permanent as it is undergoing these changes, as if it were being digitally simulated. Imagine all of its printed rules text as going gray while it is attached, as it is inactive. Then the granted Temporary keyword, because it is active, is the normal text color for Spinning Axe. There are two colors of Temporary on Spinning Axe, but the grey Temporary doesn't have any function. We can treat it as if it didn't exist for the purposes of rule 816.2.a.

816.2.Multiple instances of Temporary are redundant.816.2.a.Regardless of how many instances there are, the ability will only trigger once.

722.2. Game Effects that parse or interpret text to determine target eligibility may still parse Inactive text for the sake of eligibility. Example: Spinning Axe is a gear with [Temporary]. While it's attached and its rules text is inactive, its [Temporary] ability doesn't trigger. However, a spell that reads "Destroy a gear with [Temporary]" could still choose and destroy Spinning Axe.