  The space-borne gravitational wave interferometer, LISA, is expected to
detect signals from numerous binary white dwarfs. At small orbital separation,
rapid rotation and large tidal bulges may allow for the stellar internal
structure to be probed through such observations. Finite-size effects are
encoded in quantities like the moment of inertia ($I$), tidal Love number
(Love), and quadrupole moment ($Q$). The universal relations among them
(I-Love-Q relations) can be used to reduce the number of parameters in the
gravitational-wave templates. We here study I-Love-Q relations for more
realistic white dwarf models than used in previous studies. In particular, we
extend previous works by including (i) differential rotation and (ii) internal
temperature profiles taken from detailed stellar evolution calculations. We use
the publicly available stellar evolution code MESA to generate cooling models
of both low- and high-mass white dwarfs. We show that differential rotation
causes the I-Q relation (and similarly the Love-Q relation) to deviate from
that of constant rotation. We also find that the introduction of finite
temperatures causes the white dwarf to move along the zero-temperature mass
sequence of I-Q values, moving towards values that suggest a lower mass. We
further find that after only a few Myrs, high-mass white dwarfs are
well-described by the zero-temperature model, suggesting that the relations
with zero-temperature may be good enough in most practical cases. Low-mass,
He-core white dwarfs with thick hydrogen envelopes may undergo long periods of
H burning which sustain the stellar temperature and allow deviations from the
I-Love-Q relations for longer times.
