5°C on the suns surface would be a change of roughly 0.1%. Since radiation energy rises with the 4th power of the temperature, it would rise by about 0.4%, or by about 5 W/m².
For comparison, a doubling of the CO2 consentration in the atmosphere is the equivalent of an extra 3.7 W/m².
So yes, a long-lasting rise of 5°C on the suns surface would have significant consequences.
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