How can I fill in a large hole in a granite kitchen countertop?
The stains near the hole seem to suggest that someone had something like a coffee machine installed in this position that was plumbed for water (and maybe a small drain line) through the cabinet below. 



You could consider the possibility to place an appliance over the hole if the location and layout of your counter top was conducive to that. 

As far as patching it up.....since you are renting it is just as problematic to patch something like this as it would be going about replacing a counter top. In any case I suggest that you talk to the landlord before doing anything. They may have some ideas that would be suitable to their view of the property. Certainly it puts the landlord on notice that the hole exists and that it was not caused by you when you eventually move away. 

From a purely functional standpoint the best patch would probably be an attempt to find another piece of stone with a similar color pattern and cut out a figure 8 shaped piece to fit in the hole then bond it in place from below with another larger support piece that spans across the opening.