Place for tobogganing - that is sport involving cold, note (3,3)

I believe the answer is:

ice run

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'place for tobogganing that is sport' is the definition.
The definition suggests a singular noun which matches the answer.

'cold note' is the wordplay.
'cold' becomes 'ice' (I've seen this before).
'note' becomes 'run' (I can't justify this - if you can you should believe this answer much more).
'ice'+'run'='ICE-RUN'

'involving' acts as a link.

Can you help me to learn more?

I've seen this clue in The Independent.

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