I've seen a few people claiming:
$$\hat{H(t)}[\psi(x)T(t)] = \hat{H(t)}[\psi(x)]T(t)\tag{1}$$
i.e. an explicit function of t is not acted upon by H, even if H itself may be dependent on t.
A more specific example, Griffiths between equation 9.7 and 9.8 (implicitly):
$$\hat{H(t)}[\psi e^{iEt/\hbar}] = \hat{H(t)}[\psi] e^{iEt/\hbar}$$
Is this because t is within an exponential, or is the general statement (1) true? And why?
I feel like it has something to do with time being a parameter not a variable (although I don't fully get this concept either)