Is true altitude vertical separation between airplanes dependent on temperature?
Yes, the vertical geometric separation is temperature dependent.

If it cooler than ICAO Standard Atmosphere (ISA) conditions, the geometric vertical separation is lower than the pressure altitude separation. If the temperature is above ISA temperature then the geometric separation is higher.

The ICAO atmosphere assumes a sea level temperature of 15 degrees Celsius and a lapse rate of -6.5 degrees per 1000 geopotential meters altitude. At FL250, according to the model, the temperature is
-34.5° Celsius, at FL260 it would be -36.5°C.

Because the ISA is based on geopotential height, the geometric separation between FL250 and FL260 under standard temperature conditions would be slightly over 1000 feet. In fact it would be 1002.1 feet, because gravity is about 0.2% lower at that altitude.

If we would take the same mathematical model as in ISA, but would offset the temperature by -20° Celsius (-5°C temperature at sea level), the geometric difference between FL250 and FL260 would be only 932.7 feet.

If we offset the temperature by +20°C, the difference would be 1072.7 feet between FL250 and FL260.

This can be approximated by looking at the ratio of temperatures expressed in Kelvin. The average temperature between FL250 and FL260 is -35.5° C or 237.7 Kelvin. For 20 degrees below ISA temperature a rough estimate would yield:

$\frac{237.7 - 20}{237.7}×1000$ft = 915 feet.

For separation between aircraft and terrain, temperature effects are taken into account by adjusting the minimum safe altitude.

For vertical separation between aircraft no adjustments are made by ATC, but the effect is taken into account in safety cases.