  The observation and distinction of two compact stars with an identical mass
but a different radius would be a clear sign of hadron-quark phase transition
in nuclear matter. Motivated by studies searching for significant deviations in
the observables of twin stars, we investigate the differences that manifest in
their r-mode instability windows and spin-down evolution. Firstly, we obtain a
set of hybrid equations of state (which predict the existence of a third stable
branch of compact objects) by employing the well-known Maxwell construction
within the phenomenological framework of constant speed of sound
parametrization. Then, we systematically study the influence of certain
parameters, such as the energy density jump (in the resulting hybrid equation
of state) and the crust elasticity, on the deviations appearing in the r-mode
instability windows and spin-down evolution of twin stars. We conclude that two
stars with an identical mass and fairly similar spin frequency and temperature,
may behave differently with respect to r-modes. Thus, the future possible
detection of gravitational waves (due to unstable r-modes) from a star laying
in the stable region of the frequency-temperature plane would be a strong
indication for the existence of twin stars. Furthermore, we consider current
data for the spin frequencies and temperatures of observed pulsars and compare
them to the predictions made from equations of state employed in this study. We
find that, depending on the transition density and the rigidness of the crust,
hybrid equations of state may be a viable solution for the explanation of
existing data.
