Sorting a sorted array after increasing several elements I know that most of the efficient sort algorithms can run with a complexity of $O(n\cdot log(n))$, but this is given an unsorted array. 

However, given that the initial array is already sorted, is there an algorithm that can sort the array after multiplying several elements by 2 (or increasing them by some value) with complexity of $O(n)$?
You can update the array using the merge procedure of mergesort. Decompose your array into two smaller arrays: unchanged elements and updated elements. Since multiplying elements by $2$ preserves their relative order, you can do this decomposition in $O(n)$ while producing two sorted smaller arrays. You can then merge them in $O(n)$.