Well, Hawking was apparently assuming that growth would continue with the same trends we have today. Now this is obviously not possible, current world population is about 6.3 billion people. I did a basic glance at population growth, and actually found something reffering to a Hawking quote, something like it doubles ever 40 years. So population of 6.3 billion doubles every 40 years for 700 years...
700/40=17.5 6.3*2^(17.5)=1167791.9
Over 100 trillion. Which is a stupidly high number, but each person would not have to burn much energy to make the earth a very very warm place indeed.... hell, maybe they could do it just on body heat!