There is no problem to use LOX and CH4 at ambient temperature at the many trajectory corrections on it´s way to Mars? I guess that they will not hold cryogenic temperatures so long. I know that the cryogenic temperature increases fuel density and help cooling the engine but at space with no need of 100% engine throttle, maybe there is no need of cooling?
I think you are confusing cryogenic propellants and subcooled propellants. If they don't keep the propellants cryogenic, they will turn into a gas that would take around a thousand times as much pressure to fit in the same tanks. That would make the tanks unworkably heavy.
Raptor and Starship are designed to use subcooled propellants, which are cooled below their boiling points under 1 atmosphere of pressure, which increases their density several percent. "Warm" propellants in this context are still cryogenic, just close to or at the 1 atm boiling point (though not necessarily actually boiling if moderately pressurized).
You can't just lower the throttle to run a cryogenic liquid propellant engine off high-pressure gaseous propellants, especially one as highly optimized as Raptor. It's much more possible to allow it to run off relatively warm cryogenic propellants, though there's still going to be limits.
The actual boiling point is pressure dependent. They may carry the propellants at somewhat elevated pressure during the trip, holding the propellants at higher temperature to slow transfer of heat from the rest of the ship. They could then vent the tanks on arrival to chill the propellant by boiling it at lower pressure, accepting some losses to get it down to a temperature the engines can tolerate, whatever that turns out to be.