The thrust may well be constant, as are the V1/Vr/V2 speeds which are CAS for a given weight. Accelerate-Stop and OEI Climb Gradient performance depend upon TAS, e.g. for a RTO, stopping capability depends upon the TAS, which after considering wind component, enables us to know the speed to be dissipated in said RTO.
If the Flat rated temperature, was, for example 30 degrees at Sea Level, and V1 was 140 KIAS, TAS is 143.6 Kt. At 10 degrees temperature the TAS for the same IAS at V1 is 138.8 Kt. As the energy to be dissipated in a RTO depends upon speed SQUARED, the energy at the 30 degree takeoff is 1.07 times (107%) of that at 10 degrees. Thus, as temperature falls from 30 to 10 degrees, RTOW as limited by RTO will be 7% higher.
Similar discussion arises with respect to OEI Climb Gradient, where, for a given Rate of Climb, Gradient will be higher for a lower TAS, thus allowing greater Takeoff Weight for engines producing identical thrust.
NOTE - When using Flex or Assumed Temperature Thrust, the difference in TAS for the same speed is usually NOT considered, leaving some small conservatism. For the Limiting Weight case, it is DEFINATELY considered.
Regards,
Old Smokey