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The standard question on a knowledge test:
"Under which condition will pressure altitude be equal to true altitude?"
The answer is: "when atmospheric conditions exist, i.e., 29.92" Hg and 15 degrees C at sea level."
What I'm not understanding fully is what the temperature at sea level has to do with pressure altitude. For example, say the sea level pressure is 29.92" Hg but the temperature at sea level is 20 degrees.
Wouldn't pressure altitude still equal true altitude?
The answer in your book is a bit misleading (but mostly correct for all practical purposes). The right answer would be "when the pressure at that altitude equals the pressure you'd find at the same altitude in the standard atmosphere".
The existence of standard atmospheric conditions is sufficient but not necessary for true altitude to be equal to pressure altitude. In other words, pressure altitude and true altitude could be equal even if standard conditions are not present.
So, in your example, if sea level pressure is 29.92" Hg and the temperature is 20 degrees, pressure altitude and true altitude are equal provided that the temperature lapse rate has a higher (specific) value than the one in the standard atmosphere.
Pressure of the air above you is affected by both non-standard pressure and non-standard temperature. "From high to low, look out below" is a way of remembering that flying from high pressure to low pressure and your true altitude will decrease. It is also true with temperature.
What is interesting to me with this question is the altimeter setting is set for both non-standard pressure and non-standard temperature at the field. As soon as you increase altitude, your true altitude starts to differ from the pressure altitude read on the altimeter. I realized, though, the question doesn't assume you are at an airfield but are flying in the air.
The US AIM (7-3-1) has a great discussion on how cold weather affects pressure altitude. The FAA has also issued special procedures for airports with obstacle issues in cold weather. They are called Cold Weather Restricted Airports. You can find a current listing at the bottom of this FAA page.
https://www.faa.gov/air_traffic/flight_info/aeronav/digital_products/dtpp/search/
The pressure altitude will be equal to the true altitude (I understand this to be defined as the geopotential height above the mean sea level) when the pressure at that true altitude is equal to the pressure derived from the ICAO Standard Atmosphere model (described in ICAO Doc 7488) for that true altitude.
