Addendum.
Yes, in variable stars, variation in total energy output is usually correlated with variation in spectra and temperature; the black-body curve shifts too. However, these changes usually result from structural changes deep within the star which affect the star as a whole. The type of low-level, short-term solar variability we are familiar with now, (sunspot cycle, flares, etc) appear to be surface phenomena, and their connection to events deep inside are very poorly understood. It's sort of like trying to correlate earth's weather with events in its mantle and core.