Most ATPL level text books use the simplifying assumption that the thrust produced by jet engines is constant at all airspeeds. But in reality the thrust is maximum when TAS is zero (static thrust), then decreases as the aircraft accelerates down the runway. This is because the increasing speed of the incoming air decreases the overall acceleration given to the air as it passes through the engine. This effect can be seen in slide 10 of the link in Vila’s post 22. The magnitude of the thrust loss will increase as the OAT, and hence the TAS increases at any given EAS.
It may be the case that flat rating compensates fully for the temperature-induced loss of static thrust before brake release, but does not compensate for the thrust reduction caused by increasing TAS during climb out. If this is the case, then the thrust available during climb out will be less than the rated value and will decrease with increasing OAT even within the flat rating temperature range.