Is activation energy time-dependent?
The activation energy of a single reaction does not change with time but there might be more than one reaction happening

An activation energy is always constant for a given single reaction. But that doesn't mean the reaction speed will be constant (it might depend on concentration or temperature).

But what about autocatalytic reactions? There are two possible sources of confusion here. A very simple autocatalytic reaction might involve a changing concentration of an intermediate. In this case the reaction might go faster because that concentration is changing as the intermediate builds up (but the activation energy stays the same). In more complex cases, the catalytic route to the product involves a different reaction. What you have is not a changing activation energy for a specific reaction but the addition of a different reaction pathway to give the same product. This reaction will have a different activation energy but the activation energy of the original pathway is not changed.