So it just happened that a teammate had a remaining TP after buying BoTs so he wanted to be nice and instead of selling them to give them to me (I was supporting so he figured I could save some gold), but I had a TP in my inventory. I picked his up anyway.
What I noticed is that a shared TP's cooldown will decrease at half the time (like it's in a backpack) even though you have it in your inventory. My TP obviously had a normal cooldown decrease.

Shouldn't the shared TP also have a normal cooldown decrease instead of double (like for backpack items) ?