Do a speed test on your internet just to make sure, if both upload and download speed are okay then do this -
Right Click on Steam Icon on Windows ToolBar and then Steam->Settings->Downloads choose your region and download speed to unlimited or anything, see if it's set to a manual speed instead of auto.
If it's all normal as it should with default Steam settings, then I would say it's something related to Steam Servers with the specific game you're trying to download, it happens with me sometimes with PAYDAY 2, the download starts slowly as usual..
Hope That Helps.