For the most part, it stays at the ECN cost index. When you're running behind, they'll give you the "OT1" (on time - 1) cost index on the flight plan so you can up the climb/cruise/descent profiles.
However, there's a caveat.
At the big airlines, if you arrive too early, there's always a chance that you'll arrive at the airport and your arrival gate is still occupied, so dispatch will also print another cost index to insert in order to slow down the cruise and descent to minimize the time you spend on the tarmac waiting for gate space.
For the most part, dispatch at my airline will plan everything based off of the ECON setting. However, sometimes in the remarks section of our flight plan, it'll be denoted that it was planned on "OT1" because of enroute winds or whatever.