Suppose a pilot is just cruising around building flight time. In order to get the most flight time for fuel, would flying on a dry cold day or on a hot thermal filled day save fuel?
If the goal is simply to build time, a pilot could certainly save fuel by using thermals in a light airplane. To get an appreciable benefit, he or she will have to actually circle in the thermals, glider-style. He or she would benefit greatly by having a variometer along-- a variometer is sensitive rate-of-climb indicator designed to have minimal lag. Small portable electronic versions exist, generally designed for hang gliding and paragliding usage. One strategy would be to initially do a powered climb to near the top of the thermal convection, then reduce power to a setting that gives a slow descent and start looking for thermals to circle in to re-gain altitude without changing the power setting. However the truth is that many pilots will find circling in thermal updrafts in a light airplane to be rather fatiguing-- it involves lots of time going around in circles in bumpy air. It's a good way to expedite a climb to altitude, but it can be a tiring way to spend an entire flight.
Other options for saving fuel would include use of ridge lift or wave lift. If the aircraft is based near a long ridge, opportunities to soar the ridge at reduced power settings may be relatively common during certain times of the year when the wind direction and velocity is favorable. Using mountain wave is a more exotic proposition that may often require a long transit to the wave or a high climb to contact the wave, but I'm aware of a case where a Cessna 152 was flown in wave for over an hour with the motor at idle power (or lower) at around 6000' above ground level at a location about 15 miles from the home airport.
All things considered, the most practical way to save fuel on a regular basis is simply to operate at reduced power settings. But it certainly is technically feasible to take advantage of various forms of updrafts as well. This will take a concerted effort on the pilot's part-- it's not something that will happen automatically when the pilot is "just cruising around" on any given day, regardless of the prevailing meteorological conditions.
If the question is meant to be essentially "is it more efficient to fly in stable air or in convective conditions", the answer is "it depends". If a pilot is trying to maintain a constant altitude, then the fuel consumption will generally be lower in smooth air. If the goal is to minimize fuel consumption, the worst thing a pilot can do is to push the nose down to maintain altitude whenever the aircraft encounters an updraft. It is better to hold the airspeed constant and let the aircraft climb, and better yet to reduce the airspeed and let the aircraft climb still more. Then if desired, the airspeed can be increased to gradually shed altitude once the aircraft is out of the updraft. This strategy will reduce fuel burn during cross-country cruising flight at an average airspeed that is higher than the maximum endurance airspeed. However, if the goal is really simply to minimize the fuel consumption while accumulating hours in the air, it would be better still to simply fly at the maximum endurance airspeed in smooth air, and better still to fly at the maximum endurance airspeed on a thermally day while actually making efforts to stay in the thermals as long as possible by circling.
For flight time and saving fuel, the density is not relevant. Density is relevant to distance covered if you need a set cross country distance rather than a set time. You would only want to set power to maximum endurance which will be very near best glide CAS, while best true airspeed and ground speeds would change with density the best endurance CAS would not. It may also be advisable to plan relatively shallow descent angles so that the engine stays within a reasonably efficient power range. Climb with power low enough that you can keep it leaned.(after the initial takeoff and climb out, unless maybe you are at an excessively long desert runway) Thermals could aid the flight time allowing even lower throttle settings but could also harm as they go both up and down, capturing the right current depends on the pilot's glider experience.
Now where I rent, fuel is included in the per hour price and the goal is usually to get a set distance or set of maneuvers in a minimum time. Just the opposite of what you propose.