The complaints have been filed over the past two months, since Russia and Georgia went to war over the breakaway Georgian region on August 7.
They follow applications made by Georgia to the International Criminal Court, the International Court of Justice, and the European Court of Human Rights against Russia, accusing its neighbor of war crimes, including ethnic cleansing.
Russia has also made complaints to international courts against Georgia, and Russia's foreign minister, Sergei Lavrov, said last month Russia would help any citizens of South Ossetia wanting to make complaints against Georgia.
"There will be a massive increase in the workload of the court," Costa told Reuters. "We cannot just throw away these cases."
Asked if he thought Russia was part of a coordinated effort to overwhelm the court with applications, he said: "Yes, it's possible. It's difficult to say that it's obvious or it's likely. But it's possible."
The European Court of Human Rights also has two outstanding claims by Georgia against Russia, the first dating from 2007 and the second from the recent war. The 2007 case relates to allegations of forced expulsions of Georgians from Russia and is not expected to be completed until early next year. The other case, dealing with events in August, is only at the preliminary stage, Costa said.