Ok any cooling effect would be lost to the high exothermic reaction that is produced when you mix N2O, a hydrocarbon and an ignition source so it's not really applicable.
Oxygen (its the oxidizing agent in a combustion reaction) is not flammale on it's own and it needs a fuel source and an ignition source....just read an MSDS on it, again if it were oxygen availability then wouldn't pumping more oxygen into the engine make more sense even if its just an oxygen enriched compressed air?
It is all about thermodynamics and the stability of the end product from combustion (such as CO,CO2, H2O etc). Nitrous will immediately break down to nitrogen and an oxyanion (or more correctly radical as Dan states, my bad) as nitrogen prefers to be on it's own (more stable), the oxygen free radical can now have a few options 1.) It can rebond to a nitrogen molecule to reform nitrous 2.) It can rebond with another oxygen molecule (it is not stable bonding to just another radical to form oxygen as the quantum physics dont add up from this reaction) to form ozone or 3.) It can oxidize (donating the radical electron) the hydrocarbons producing an enormous amount of energy and forming very stable products such as water and CO.
If one was to look up the gibbs free energies of these reactions they would find that option number three will be the most favoured both thermodynamically, and from a stoichemetry point of view. Anyway this translates into more energy released per unit fuel.
N2O + Hydrocarbon + e- ----> N2 + H2O +CO + Heat
It really isn't possible to have pure oxygen produced by reacting two moles of N2O with the prescence of fuel, a spark, pressure and higher temp its just not thermodynamics or quantum physics.
This is a basic explanation off the top of my head (and now it hurts...hahaha) so please don't make me look up my 3rd year uni Physical Chemistry Books and give you an actual numbers.
"Inflammable means flammable......What a country!"