  To the best of our knowledge, there are no specific calculations of
gravity-darkening exponents for white dwarfs in the literature. On the other
hand, the number of known eclipsing binaries whose components are tidally
and/or rotationally distorted white dwarfs is increasing year on year.
  Our main objective is to present the first theoretical approaches to the
problem of the distribution of temperatures on the surfaces of compact stars
distorted by rotation and/or tides in order to compare with relevant
observational data.
  We find discrepancies between the gravity-darkening exponents calculated with
our methods and the predictions of the von Zeipel theorem, particularly in the
cases of cold white dwarfs; although the discrepancy also applies to higher
effective temperatures under determined physical conditions. We find physical
connections between the gravity-darkening exponents calculated using our
modified method of triangles strategy with the convective efficiency (defined
here as the ratio of the convective to the total flux). A connection between
the entropy and the gravity-darkening coefficients is also found: variations of
the former cause changes in the way the temperature is distributed on distorted
stellar surfaces.
  On the other hand, we have generalised the von Zeipel theorem for the case of
hot white dwarfs. Such a generalisation allows us to predict that, under
certain circumstances, the value of the gravity-darkening exponent may be
smaller than 1.0, even in the case of high effective temperatures.
