Will this work:

SELECT A.Customer

, A.[Status]

, NumDays = SUM(DATEDIFF(d,A.LastUpdateDate , COALESCE(B.LastUpdateDate, GETDATE())))

FROM #TempTable A

OUTER APPLY (SELECT TOP 1 LastUpdateDate FROM #TempTable

WHERE LastUpdateDate > A.LastUpdateDate

AND A.Customer = Customer

ORDER BY LastUpdateDate) B

Group...