For simplicity we're going to ignore aerodynamics and vehicle mass for a moment...

Are small (around 1 litre) car engines designed to tolerate a higher duty cycle than large (2+ litre) car engines? 

Cars are usually all travelling at roughly the same speed which means the engines will be putting out proportionally similar amount of power, however a small engine would be operating at a greater proportion of its output than a large one to maintain the same speed. 

For some background to this, I have a 1.1 l car and I spend quite a lot of time with my foot completely flat on the floor, especially going uphill on a high speed road (60 or 70 mph limit) to keep up with the traffic. I think my record's somewhere around 2 minutes on full throttle.