If I give an equipment temporary, does it die?

It depends on how the equipment gained Temporary.

• If Temporary was printed on the equipment (e.g., Spinning Axe): No, it does not die while attached. When equipment is attached, its printed rules text becomes Inactive (Rules 135.4, 718.2), so printed Temporary cannot trigger.

• If Temporary was granted by an external effect (e.g., Turn to Dust): Yes, it dies. Granted rules text remains active even while the equipment is attached. That active Temporary will trigger at the start of its controller's next Beginning Phase and kill it.

Same goes if the equipment already has printed Temporary and gets a second copy granted — the granted one stays active and will kill it.