OK, Vr-V2 speed spread will increase on a colder day, because the thrust will be higher (unless you're totally 'flat rated') which will provide more acceleration at a given weight.
Whether one or both of Vr and V2 are constant depends on what the limiting factors are for either.
Vr may be driven by V1, in which case the limiting factors on V1min may be relevant. V2 may be driven by Vmca OR Vs(Vsr). Without knowing the details for the specific conditions, it's impossible to know what the impact on Vr or V2 will be absolutely, but the relative behaviour is thrust-driven, as noted in the first para.
And V2 isn't single engine minimum control speed; that's Vmc (a, g or l, as approrpiate to the phase being considered)