Perhaps the biggest difference between some turbojet equipment and piston or turboprop equipment is the range and the capabilities. Many turbojet aircraft tend to go higher (although we used to do FL410 in the Piaggio Avanti, and it was faster than a Cessna Citation). Crossing large areas requires a careful look at the weather, overflight issues, and evolving depressurization issues.
I'm not sure what the significance of 10,000' is, given that one may be over the ocean on one portion of a leg, and over high terrain exceeding 18,000' on another. Look at the evolving terrain beneath the aircraft and make decisions...not just a line at 10,000' (which becomes irrelevant if higher terrain is below).
Knowing where one's options are at all times makes a transition from a normal point-to-point flight much easier in the event of a diversion. Knowing one's fuel state in relation to one's diversion capabilities is important. That is, if one has to descend in the event of a depressurization or during a fire and subsequent depressurization, what are one's options at any given time with the fuel available, considering terrain, and weather? What was true and hour ago may not be true now, particularly given evolving weather and changing terrain beneath the aircraft.
Turbojet aircraft are used because they're fast and they can cover long distances. This may mean overflight issues; one may have to consider the ramifications of needing to detour around a country rather than overfly it, either during a diversion, or in the normal course of operations. This can affect fuel reserves and mandate changes in the flight. A diversion while going into Afghanistan, for example, may mean that while a field in Iran is closer, it's not advised...have this in mind before reaching a point where things may become critical, and have options always in mind.
The optimum flight altitude changes during the flight as weight is lost, and ensuring the most efficient operation of the airplane is an ongoing evaluation. One must look not only at the optimum altitude for fuel consumption vs. cruise speed, but also headwinds and weather.
Enroute, navigation is a primary concern, of course, and depending where one is, maintaining various types of communications, position reports, etc, is also part of the process. Position reports, in turn, sometimes involve additional information such as ongoing updates or estimated time of arrival, accurate fuel updates,and so forth. This information is passed on to the aircraft operator (airline or company), and used to make determinations regarding diversions, re-release points, and so forth.