  Certain relations among neutron-star observables that are insensitive to the
equation of state are known to exist. Such universal relations have been shown
to be valid for cold and stationary neutron stars. Here, we study these
relations in more dynamic scenarios: protoneutron stars and hypermassive
neutron stars. First, we study protoneutron stars. We use an effective equation
of state, extracted from three-dimensional core-collapse supernova simulations,
to obtain the structure of spherically symmetric protoneutron stars. We then
consider nonradial oscillations to compute their $f$-mode frequency ($f$), as
well as slow rotation and small tidal deformation, to compute their moment of
inertia ($I$), spin-induced quadrupole moment ($Q$), and Love number. We find
that well-established universal relations for cold neutron stars involving
these observables ($I$-Love-$Q$ and $f$-Love relations) are approximately valid
for protoneutron stars, with a deviation below $\approx$ 10$\%$ for a
postbounce time above $\approx$ 0.5 s, considering eight different supernova
progenitors and the SFHo equation of state. Next, we study hypermassive neutron
stars. We obtain a new universal relation between the $f$-mode frequency and
the compactness of cold and nonrotating neutron stars, using bulk quantities.
We show that this relation has an equation-of-state-variation of $\approx$
$3\%$, considering a set of ten equations of state. Using results from binary
neutron star merger simulations, we study the evolution of hypermassive neutron
stars on the $f$-$C$ plane, considering two different mass ratios and the SFHo
equation of state. We find that the relation between the peak frequency of the
gravitational-wave signal and the compactness from these hypermassive neutron
stars deviates from the universal $f$-$C$ relation by 70 $-$ 80$\%$, when the
peak frequency is taken directly as a proxy for the $f$-mode.
