Which precautions should I take when flying in very cold / hot weather?
Pre-heat the engine
When the engine is cold, the metals may come in contact to each other, which greatly increases engine wear. The best way to pre-heat is to use a hangar.
Engine oil
Oil viscosity is closely related to temperature. If the temperature is too low, oil becomes too thick. Thick oil cannot flow around the engine compartments freely. Thin oil, on the other hand, may have reduced lubricating effect. Thus, choosing the correct oil type is important. An oil with lower viscosity may be used in cold climate, and likewise oil with high viscosity in a hot climate.
Cabin heat
Cabin heat can be useful for warming the cabin, however a leakage may lead to carbon monoxide poisoning. Carbon monoxide detectors can be useful in providing an early warning.
De-ice
Ice built up can affect aerodynamic surfaces severely. In serious cases, they may cause the wings to lose enough lift and stall. Pitot tubes and engine sensors may also give false readings if their de-ice/anti-ice system is not activated.
Density altitude
The density of cold air is greater than normal. Engine damage may be induced by pushing full throttle, which causes the engine to produce more power than it is rated.
In hot air, the density is reduced. This reduces engine performance, resulting in longer takeoff and landing rolls. Aerodynamic performance is also reduced, resulting in a higher TAS (true airspeed) for the same IAS (indicated airspeed). These effects must be taken into consideration in determining whether runway length is sufficient.
If there is high terrain near an hot airport, it may be necessary to circle the airport to climb as the plane may not have sufficient climb performance.
Surface friction
In cold conditions, the taxiways and runways are often icy and slippery. Heavy braking should be avoided.