In the Denver ISA case, the engine probably is "altitude bumped" (N1 or EPR) to create approximately the same thrust as at SL - That's usually the case with airliner engine ratings. (Disclaimer - I cannot speak for GA turbine engines!)
Thus at same TOGW, the aircraft acceleration (Kts GS/sec) should be about the same. (A = F/M and all that) However - to reach V1, Vr etc. (IAS) the ground speed will be higher, thus the time (and runway) to rotate is longer!
Coming down to the Phoenix case, engine thrust (N1 or EPR) will be cut back for hot section protection in the high OAT. Thus aircraft acceleration is reduced, given the same TOGW. But - Ground Speed for a given V1, Vr etc. is higher than SL because of the TAS/IAS relationship at high OAT.
Thus high GS, and reduced acceleration, gives an even longer runway requirement at PHX.
These are general principles only, and different engine rating methods may affect the results. However, I believe most airliners will follow these guidelines.