Sounds like a worn bearing. I'd go waaaay out on a limb and theorize that while the engine is warming the tolerances of the component making the sound are changing due to metal thermal expansion.

Agree with Mathew1 on bearing. My guess is water pump vs timing belt tensioner pulley. 2 possible ways i would adress this issue depending on my day and preference. 1: Agressive approach: remove all 3 drive belts while engine is cold and manually turn each component to feel if any bearing/component ...