It's impossible to be 4 times as efficient as an internal combustion engine unless you are talking averages, including idling.  SOFCs are around 60% efficient.  The Prius powerplant hits about 38%, so the improvement at peak efficiency is closer to 50%.

That said, this cell has great possibilities.  The high operating temperature of 750°C is considerably hotter than gas-turbine compressor exhaust air, allowing that air to be used for SOFC air supply and cooling.  That in turn allows the SOFC to function as a topping cycle for the gas turbine, with the SOFC waste heat doing work a second time in the GT.  Operating under pressure, the reactant concentrations in the SOFC will be marginally higher and so will the operating voltage; this may increase efficiency slightly.

If we assume that an SOFC unit and afterburner replaces the combustor on a GE LMS100 gas turbine, and that the turbine requires approximately 50 MW(th) to idle (projecting its operating curve to zero power suggests 30-odd MW, but that is probably optimistic), the SOFC operating at 60% efficiency would output 75MW(e) and 50 MW(th).  Adding methane to the SOFC exhaust would add thermal power.  Bringing the LMS100 up to its full 217.4 MW(th) heat input would add 167.4 MW(th) of fuel and add 100 MW(e) of output.  The net efficiency at full power would be 175 MW(e)/292.4 MW(th) = 59.9%.  Not too shabby.

There are other ways to increase the system efficiency than using just waste heat in a combined cycle. By combusting the tail gas (the SOFC only needs the h2) a hybrid SOFC turbine or hybrid SOFC IC engine can achieve greater than 70% efficiency. Check out this Stanford reference (http://gcep.stanford.edu/events/symposium2013/docs/Poster2013_RebeccaPass.pdf).
General Electric is also investigating using SOFC and their Gas Engines (see http://spectrum.ieee.org/energywise/energy/fossil-fuels/ge-to-muscle-into-fuel-cells-with-hybrid-system, also patent US 20120251899, SOFC high-efficiency reform-and-recirculate system).

gryf, SOFCs can run on pure carbon monoxide.  They are oxygen-ion transport cells, and any fuel that scavenges oxygen at the anode will do.

What difference would it make if end users had to pay for CO2 emitted @ $40/tonne to $100/tonne?

A practical point is that the relatively pure oxygen of the stratosphere would benefit the SOFC, and the waste heat would be invaluable for deicing in flight and before takeoff, if the battery system were distributed across the wings and a fluid based heating system were manageable. There is still a great need to cool the jets, even from runway heat, so coolant management will always be there. The big issue is the energy cost and benefit of compressing the air at high altitudes to run the SOFC. What pressure range does the model here operate at? Cabin air pressure and power to maintain it are ongoing issues. So is CO2 and particulate scrubbing, for which the basic tech here is a help.

Engineer-Poet is correct, when hydrogen is derived by “reforming” hydrocarbons such as gasoline or dodecane in the presence of limited oxygen, the “reformate” gas includes CO which is converted to CO2 at the anode. Since the reforming process does not convert 100% of the fuel or have a 100% H2 yield, the combustibles content of the tail gas can vary widely.

gryf, you're mistaken about SOFCs in general.  The only fuel cells that require hydrogen are PEM (Proton Exchange Membrane) FC's.  Hydrogen supplies the protons which carry charge across the electrolyte.  SOFCs use oxygen ions, which are supplied from the air.  They need no hydrogen at all.

SOFCs generally have combustible tail gas because it's not efficient to completely consume the fuel; as the fuel goes to complete consumption the oxygen content in the gas rises and the cell voltage drops.  Leaving enough un-consumed fuel to burn with some excess air cleans up the emissions.

There's potential for an SOFC APU, but the real advantage would come from a full integration of SOFCs with the main engines.  If the total engine efficiency could be boosted from 40% to 60%, fuel consumption would drop by a third; electrification of much of the propulsion system would allow further improvements from e.g. boundary-layer control.  That is an immense advantage.

The hybrid FC/turbine engine would use the FC as an air pre-heater for the gas turbine.  Two bites at the energy apple.

Good point, Simon. However, a diesel is not an option for jetliner. Diesel can do about 45% peak thermal efficiency, while a more recent turbofan core engine with pressure ratio of 40 or greater can deliver 45-50% thermal efficiency at cruise, depending on size and pressure ratio and turbine inlet temperature. It is very hard to improve on this kind of efficiency and still keep the core engine light and compact enough for a jetliner.

It's hard to use a diesel as a topping cycle for a gas turbine.

If we assume a 500°C fuel-cell air inlet temperature and 800°C output temperature, the air can carry away about 302 kJ/kg of heat.  A 40%-efficient SOFC would dump that much heat while generating 201 kJ(e)/kg of air.  I'm not sure how much heat-dissipation capacity to allocate to the fuel, but using the FC to pre-heat and reform the fuel would allow more heat dissipation in the FC and increase the amount of fuel energy recoverable as electricity.

IIRC, the best modern gas turbines have turbine inlet temperature limits on the order of 1380°C, a temperature rise of about 880°C over the compressor outlet.  The FC would produce about a third of this ΔT.  The actual fuel energy input to the FC would be 5/9 of the total turbine heat input, with FC electric output of 2/9 of the turbine heat input.  If the GT efficiency remains at 40% of heat input, the net efficiency becomes:

2/9 / 11/9 = 2/11 of fuel energy recovered in the FC (18.2%)
9/9 / 11/9 * .40 = 32.7% recovered in the GT

Net efficiency 50.9%, up more than 25% over the bare GT.