Or would the cost to the airline be about the same as fitting GE engines?
There isn't another RR engine that would perform the A-380 task, but there is an engine (partly made by GE & partly made by P&WA) on some A-380's, the GP7200. Emirates and Air France's A380's have been in service for sometime using GP7200's.
From an Airbus ceremony in Toulouse, France, December 2010:
“The GP7200 engine has been performing very well on the A380,” said Alain Flourens, Airbus Executive Vice President, A380 Program. “We congratulate the Engine Alliance on its 100th Engine milestone and look forward to many more engine deliveries to come.” The GP7200 powers the Airbus A380 aircraft, and the 100th engine is destined for Korean Air's first A380. The airline is expected to take delivery in May 2011. There are 19 GP7200-powered A380s in service. EA launch customer Emirates began operating the aircraft in August 2008 and received its fifteenth A380 on November 30. Emirates is Airbus' largest A380 customer with orders for 90 of the super-jumbo aircraft. Air France entered service with the GP7200-powered A380 in late 2009 and has four A380s in its fleet and eight more on order. Other GP7200 customers, in addition to Korean Air, include Etihad Airways, Air Austral and International Lease Finance Corporation (ILFC).