Obviously higher temperature than standard and lower pressure than standard (high density altitude) hurt takeoff performance. The wings have less air molecules to produce lift with, and a typical piston engine will produce less power. But does this translate to less range/fuel endurance in cruise? The performance charts of the PA-28 I fly seem to indicate the opposite.
2 Answers
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As you go to higher altitude and the density drops, you end up flying at higher true airspeed to compensate. Namely, you want to fly at about the same lift coefficient (and thereby the same L/D) independent of altitude. Since your weight is the same, and $C_L=W/(q S)$, you hold constant dynamic pressure (q).
$q=0.5\rho V^2$ Flying constant dynamic pressure works out to the exact same thing as flying at constant equivalent airspeed -- KEAS.
However, holding constant KEAS with increasing altitude means you're flying at higher KTAS.
The power required for flight will increase -- $P=D V$. The drag will be constant, but the velocity (here true) is increasing. Depending on whether you have a fixed pitch or constant speed prop and where you are in its operation, your propeller efficiency might increase or decrease a bit. Your BSFC might also increase or decrease slightly depending on details of the required throttle setting.
The 'getting there faster' and the 'burning more fuel per hour' cancel out and you end up with pretty much constant specific range (miles per gallon) for a piston prop aircraft (when varying altitude).
However, humans value getting someplace faster, so it is preferred to fly higher. In fact, aircraft are often operated at slightly faster than the speed for best range -- the extra fuel required is small, but it can come with meaningful time savings.
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Both higher temperature and lower pressure mean lower air density. This will allow the plane to fly faster at the same polar point. The higher temperature will hurt engine efficiency a bit, but the effect on flight speed will outweigh this, so a net gain in true air speed versus fuel flow could remain, depending on propeller characteristics.
Reasoning: To fly at the same polar point, dynamic pressure must be the same. Lower density needs higher speed to keep dynamic pressure constant, so true air speed goes up with the reduction of the square root of density.
The efficiency of heat machines is proportional to the ratio between the temperature difference within the cycle divided by the maximum absolute temparature. However, given the wide spread between air temperature and the highest temperature during combustion, the change in true air speed is larger, so the reduction in efficiency will be smaller than the gain in true air speed. Since your measure of performance is range in cruise (ground covered for fuel burnt), a higher true air speed at the same fuel burn will directly increase range.
Details now depend on your propeller efficiency over true air speed: For a fixed pitch prop it could go up or down, depending on where the speed for best prop efficiency is relative to flight speed. For a constant speed prop efficiency drops, basically eating up the gain in fuel flow over speed.
