Could adding a gearbox to the Trent 900 improve the efficiency of the A380?
It would not be a simple task to add a gearbox to an existing engine like the Trent 900. The whole engine is designed to operate as a whole, with each of the three shafts rotating at an optimal RPM. The gearbox would have to be designed to work with the existing engine hardware, which would mean compromises on efficiency. The engine would also have to be modified to find room to fit the gearbox in.

A gearbox also does not automatically mean an engine will be more efficient. The geared PW1000G achieves similar efficiency to the CFM LEAP which does not have a gearbox.

It's much more likely that a new engine would be designed for the A380. Airbus has previously said that this was planned for some point in the future. However, a new engine is not cheap for either the engine company to develop or for Airbus to integrate. The A330 and A320 programs have gone this route but they are also designs with high demand.

A new engine would have to generate enough demand to justify the cost, and it does not seem that Airbus sees this demand yet. Airbus has been struggling to find new orders for the A380 and has been slowing down production. It's not clear that better efficiency is enough to make the plane sell. Four engine planes are inherently limited in their efficiency versus twin engine planes. The four engines add more drag and weight versus two engines, and increase maintenance costs.

The A380 also requires special accommodation in the air and on the ground for its large size. The 747 has been around for much longer and does not need as many special accommodations. Boeing went to the expense of redesigning the wing and adding new engines for improved efficiency, yet is still struggling to find orders.

More recently Airbus has announced that they are considering "A380 plus" which will include new winglets, wing changes, and other design tweaks to improve efficiency. This will be a much cheaper way to add efficiency than a new engine, if that efficiency will mean more orders.