Temperature can be a factor. I have seen two turbine engines where OAT had to be taken into consideration. The -9 and -9A versions of the JT8D had different max start temps depending on whether the OAT was above or below 15°C. The Dart engine, at least on the aircraft I flew, had different fuel trimmer settings for start and takeoff depending on OAT.
But I have never heard ATC giving the temperature to pilots as what would appear to be a policy when pushback or engine start was requested.