  Using the formalism of general relativity for axially symmetric gravitational
fields and their sources - rotating compact stars - a perturbation theory with
respect to angular velocity is developed and physical quantities such as mass,
shape, momentum of inertia and total energy of the star are defined. The change
of the internal structure of the star due to rotation has been investigated and
the different contributions to the moment of inertia have been evaluated
separately. Numerical solutions have been performed using a two-flavor model
equation of state describing the deconfinement phase transition as constrained
by the conservation of total baryon number and electric charge. During the spin
down evolution of the rotating neutron star, below critical values of angular
velocity a quark matter core can appear which might be detected as a
characteristic signal in the pulsar timing. Within the spin-down scenario due
to magnetic dipole radiation it is shown that the deviation of the braking
index from $n=3$ could signal not only the occurrence but also the size of a
quark core in the pulsar. A new scenario is proposed where, due to mass
accretion onto the rapidly rotating compact star, a spin-down to spin-up
transition might signal a deconfinement transition in its interior.
