  Neutron stars are the densest objects in the Universe, with $M \sim 1.4
M_{\odot}$ and $R \sim 12$ km, and the equation of state associated to their
internal composition is still unknown. The extreme conditions to which matter
is subjected inside neutron stars could lead to a phase transition in their
inner cores, giving rise to a hybrid compact object. The observation of
$2M_{\odot}$ binary pulsars (PSR~J1614-2230, PSR~J0343$+$0432 and
PSR~J0740$+$6620) strongly constraints theoretical models of the equation of
state. Moreover, the detection of gravitational waves emitted during the binary
neutron star merger, GW170817, and its electromagnetic counterpart, GRB170817A,
impose additional constraints on the tidal deformability. In this work, we
investigate hybrid stars with sequential phase transitions hadron-quark-quark
in their cores. We assume that both phase transitions are sharp and analyse the
rapid and slow phase conversion scenarios. For the outer core, we use modern
hadronic equations of state. For the inner core we employ the constant speed of
sound parametrization for quark matter. We analyze more than 3000 hybrid
equations of state, taking into account the recent observational constraints
from neutron stars. The effects of hadron-quark-quark phase transitions on the
normal oscillation modes $f$ and $g$, are studied under the Cowling
relativistic approximation. Our results show that, in the slow conversion
regime, a second quark-quark phase transition gives rise to a new $g_2$~mode.
We discuss the observational implications of our results associated to the
gravitational waves detection and the possibility of detecting hints of
sequential phase transitions and the associated $g_2$~mode.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology