Why is the Sun's core power density only 276 W/m³ if my calculation gives 3.42×10^7 W/m³?

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It appears I will not get an answer to my question, so for the benefit of the OPer, I will answer it myself: if you replace fusion in the Sun by a similar process that releases energy twice as fast at the same temperature and density, it will not have any significant impact on the luminosity of the Sun. This is obvious, is it not, from the well-known fact that the Sun's luminosity did not even change when fusion first initiated.
 
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In one of Brian Cox's popular science broadcasts, he relates a history related to the sun's modest toaster oven power density. IIRC, he stated when the first estimates of the sun's radiated power were calculated (from the time to melt ice via sunlight on the Earth and the sun's distance was known), the results led some to the (seemingly wacky) guess that the sun was burning something like coal, as the total radiated power was a rough match for a coal fired body of that size. That coal would have been consumed too quickly did not immediately squash the hypothesis, since at the time a ~10,000 year old Sun/Earth was still fashionable.