Yaris starts then it wont start.

[slow.yaris]

Wouldn't fuel pump, spark plugs, sensors all throw a code? I would check for codes even if its not showing up, the LED could be bad, or previous owner could have taken it out.

I understand the warming and cooling down thing, but I think it would be really weird to have a fuel pump or spark plugs that won't START when warm, but give you no trouble when driving around - while warm.

Ignition coils is my guess, but I also agree, take it to a mechanic before you start changing random parts.