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No, they don't. Temperature and size aren't really all that related.

Color and temperature are related. Supergiants come in all colors from red (relatively cool) to blue (very, very hot). Note that even "cool" here is a few thousand Kelvin (aka "red hot").

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The surface temperature of most super giant stars is usually low (they appear red) however their total energy output is extremely high (it is just spread over an enormous surface area).

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