I assume that the house edge is similar for each sport, regardless of the sportsbook. I will use the same event for each sport to make the comparison fair. I am aware that some sportsbooks may offer better odds for certain leagues or sports, so I will use multiple sports to balance that out.
If I'm not wrong, then there is really difference between the house edge of different sports event in a sportsbook. Some sportsbook tries to give the best odds for the popular league, where you will notice a big difference on the unpopular league odds and house edge.
This is a lot of work, but it may interest some people. However, I doubt that anyone would switch sportsbooks unless they find a huge difference in the odds.
Sports bettors always search for better odds with higher betting limit. Some bettors will obviously switch the sportsbook if the notice any reputable betting site with a better odds.