Gas Turbine Start-up Time Difference of Steam and Air Cooled One

The steam-cooled system is much more complex than you describe, and requires paying close attention to clearances, rate of temperature change, and differential expansion to name a few.

The auxiliary boiler that you refer to is actually a HRSG (Heat Recovery Steam Generator) whose input energy is derived from the turbine's exhaust gas stream, with a portion of its output steam used to cool the turbine, hence there is a long heat-soak sequence that involves multiple control loops and much more metal that has to be brought into equilibrium before the unit can begin to accept load.

In contrast an air-cooled turbine is basically a once-through system with cooling air bled off of the compressor stage(s). Spin it up, inject fuel, and it reaches operating temperature ready to accept load much faster.

For further insight into these complexities, read the description from this patent.

You may have quite a different concept of what you are talking about, from what I know. The sizeable gas turbine emergency backup generators are up and on-line in second(s). Like airplane engines, what they really are.

Care to elaborate?

Yes, another difference between the commercial world and the utility world. You are thinking jet engine based generation, OP is referring to a specialized form of the industrial gas turbine, two entirely different forms of doing the same thing.

Jet engine based generation does not have a direct mechanical connection to the generator, instead the exhaust gas is passed over a free turbine that drives the generator directly. They can be started and online in less than a minute but have the poorest heat rate (they cost a lot of fuel to run) of any form of generation.

Industrial gas turbines, on the other hand, take their power off the shaft, or through a gear box, that drives the generator directly. The exhaust gases can then passed through a HRSG for additional gains in efficiency, especially when used in a combined cycle power plant. These systems take a much longer time to start up (but considerably less than a base station), and can have heat rates that make them very attractive for constant use.