  We construct equilibrium models of uniformly and differentially rotating
hybrid hadron-quark stars using equations of state (EOSs) with a first-order
phase transition that gives rise to a third family of compact objects. We find
that the ratio of the maximum possible mass of uniformly rotating
configurations - the supramassive limit - to the Tolman-Oppenheimer-Volkoff
(TOV) limit mass is not EOS-independent, and is between 1.15 and 1.31,in
contrast with the value of 1.20 previously found for hadronic EOSs. Therefore,
some of the constraints placed on the EOS from the observation of the
gravitational wave event GW170817 do not apply to hadron-quark EOSs. However,
the supramassive limit mass for the family of EOSs we treat is consistent with
limits set by GW170817, strengthening the possibility of interpreting GW170817
with a hybrid hadron-quark EOSs. We also find that along constant angular
momentum sequences of uniformly rotating stars, the third family maximum and
minimum mass models satisfy approximate EOS-independent relations, and the
supramassive limit of the third family is approximately 16.5 % larger than the
third family TOV limit. For differentially rotating spheroidal stars, we find
that a lower-limit on the maximum supportable rest mass is 123 % more than the
TOV limit rest mass. Finally, we verify that the recently discovered universal
relations relating angular momentum, rest mass and gravitational mass for
turning-point models hold for hybrid hadron-quark EOSs when uniform rotation is
considered, but have a clear dependence on the degree of differential rotation.
