if you are talking about energy, that energy is generated far earlier than the formation of the sun itself.
in fact, it has been there forever, like in the beginning of the universe. Why do you suddenly ask about its "lifetime".

plus, energy doesn't have to propagate fast, and the propagation speed has nothing to do with the speed of light. In fact, that is an upper limit. energy doesn't propagate that fast most of the time, unless it happens to be in the form of a photon and travel in the vacuum.

Why do we even discuss it anyway? I remember someone was even surprised that "speed of light " is so slow in side the sun. Isn't it just bullshit?