Star's cooler patch on a tropical beach? (7)

I believe the answer is:

sunspot

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'a tropical beach?' is the definition.

'star's cooler patch on' is the wordplay.
'star' becomes 'sun' (Sun is an example).
'cooler patch' becomes 'spot' ('spot' can be a synonym of 'patch'. I am not sure about the 'cooler' bit.**).
'on' means one lot of letters go next to another.
'sun'+'spot'='SUNSPOT'

Can you help me to learn more?

(Other definitions for sunspot that I've seen before include "Solar dark patch" , "cause of magnetic storm" , "warm place" , "Cool darker patch on our star" , "Dark cool patch on the surface of our nearest star" .)

I've seen this clue in The Telegraph.

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