Doubling the ingredient of Japanese cheesecake: more time to cook?
It's really hard to say, there are many factors to consider: oven size and power, the recipe ingredients, pan size and shape, etc. The recipe calls for an 8" or 9" round pan, I will use an 8" pan in calculations. The main question is area, an 8" pan's area is 50 square inches (that's pi times the radius squared), while the area of a 10 x 10 springform pan is 100 square inches, double that of the 8" pan. Your cheesecake will actually be the same thickness as the pan the recipe calls for, so the the baking time is not going to be dramatically different. It will be a bit longer because the cake area is bigger, that could be just a few extra minutes. Keep in mind that recipes can often be way off in baking times. Other Japanese Cheesecake recipes call for baking an hour, not 30 minutes, for example. 

The recipe does not say how to check your cake is done, which is not good. Baking times vary according to many factors, and you should always bake for a result using time as a guideline. Most recipes say to bake until it's a golden brown as your first guideline, given it's actually a souffle I'd use the touch test, ie push it down with a finger, if it springs back up it's done.

My advice would be to make this recipe smaller scale before doubling it, make sure you get the ins and outs right before you make it for dinner guests.