Presumably, it would be expensive to use ozone ($\ce{O3}$) as an oxidizer instead of $\ce{O2}$, but would the extra oomph be worth it? Does $\ce{O2}$ provide as much thrust/energy/heat as can be provided, given the liquid hydrogen propellant?
I figured Chemistry was better than Physics, Engineering or Astronomy for this question.
As is usual with rocket fuels, the problems of ozone are practicality not performance
Almost every answer for why a specific rocket fuel component is used or not will end up referring to John D Clarke's magnificent and sparklingly written book: Ignition: An informal history of liquid rocket propellants (a rare technical book worth reading for the brilliant and humorous style in addition to the technical content).
His summary of why ozone, despite its apparent functional advantages, is not more widely used is fairly simple: the practical problems outweigh the apparent advantages.
He points out the advantages:
What makes it attractive as a propellant is that (1) its liquid density is considerably higher than that of liquid oxygen, and (2) when a mole of it decomposes to oxygen during combustion it gives off 34 kilocalories of energy, which will boost your performance correspondingly. Sänger was interested in it in the 30’s, and the interest has endured to the present. In the face of considerable disillusionment.
But every available way of creating a liquid with a high proportion of ozone is dangerous. Ozone is extremely toxic and unstable:
For it has its drawbacks. The least of these is that it’s at least as toxic as fluorine. ... Much more important is the fact that it’s unstable—murderously so. At the slightest provocation and sometimes for no apparent reason, it may revert explosively to oxygen. And this reversion is catalyzed by water, chlorine, metal oxides, alkalis—and by, apparently, certain substances which have not been identified. Compared to ozone, hydrogen peroxide has the sensitivity of a heavyweight wrestler.
Work was done on solutions of ozone in liquid oxygen which is somehow more stable. But has the disadvantage that ozone/oxygen mixtures separate into two phases the ozone rich version of which is difficult to prevent in feed tubes after firing and is extremely unstable. Another mixture considered to make handling easier was with liquid fluorine (!!!).
Ultimately he concludes the known work on ozone mixtures of any sort:
For ozone still explodes. Some investigators believe that the explosions are initiated by traces of organic peroxides in the stuff, which come from traces, say, of oil in the oxygen it was made of. Other workers are convinced that it’s just the nature of ozone to explode, and still others are sure that original sin has something to do with it. So although ozone research has been continuing in a desultory fashion, there are very few true believers left, who are still convinced that ozone will somehow, someday, come into its own. I’m not one of them.
Maybe there are theoretical advantages, but they are outweighed by the practical and safety problems. In a profession used to testing things like FOOF (fluorine dioxide) and chlorine trifluoride, this is some admission to make.
Liquid ozone is rather stable (as in "can be stored") as an up to 70% solution in LOx, if you add some stabilisers. That would give quite a bit of additional boost in a rocket, but it's surely going to explode already in the rocket's fuel pumps and tubing, instead of burning in the engine's nozzle.
Rocket engines are tricky enough to build, you don't want to add the requirement that all material is of a kind that does not catalyse a premature decomposition of ozone. It's usually a runaway process: Decomposition increases temperature, temperature increases decomposition rate, .... boom.
