I do feel some sarcasm here, but anyway..Wait wait, I would not listen to the amp immediately after turn-on! The electrons in the tubes need about 15 minutes to stabilize-- listening sooner can actually cause damage to the tubes. It's not like with solid state stuff where you can literally turn on and start listening cold. Be careful!
1) The muting relay won’t even connect the output until everything has stabilized to the point that it is safe to do so, and 2) tubes won’t be damaged by being connected in circuit cold - they just won’t conduct until the cathode is hot enough to let off electrons (however, even with an indirectly heated cathode that’s just seconds in most cases).
That said, the characteristics of all electronic components will drift a little bit with temperature until they have reached some sort of thermal equilibrium. Does that matter? Maybe. Maybe not. Depends on the usage.