A tempo that's fast (5)

I believe the answer is:

apace

'fast' is the definition.
(apace can mean quickly)

'a tempo' is the wordplay.
'tempo' becomes 'pace' (synonyms).
'a'+'pace'='APACE'

'that's' is the link.

(Other definitions for apace that I've seen before include "at high speed" , "At a rapid rate" , "Quick" , "What's quickly" , "flat out" .)

I've seen this clue in the Evening Standard.

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