  Relations between neutron star properties that do not depend on the nuclear
equation of state offer insights on neutron star physics and have practical
applications in data analysis. Such relations are obtained by fitting to a
range of phenomenological or nuclear physics equation of state models, each of
which may have varying degrees of accuracy. In this study we revisit
commonly-used relations and re-assess them with a very flexible set of
phenomenological nonparametric equation of state models that are based on
Gaussian Processes. Our models correspond to two sets: equations of state which
mimic hadronic models, and equations of state with rapidly changing behavior
that resemble phase transitions. We quantify the accuracy of relations under
both sets and discuss their applicability with respect to expected upcoming
statistical uncertainties of astrophysical observations. We further propose a
goodness-of-fit metric which provides an estimate for the systematic error
introduced by using the relation to model a certain equation-of-state set.
Overall, the nonparametric distribution is more poorly fit with existing
relations, with the I-Love-Q relations retaining the highest degree of
universality. Fits degrade for relations involving the tidal deformability,
such as the Binary-Love and compactness-Love relations, and when introducing
phase transition phenomenology. For most relations, systematic errors are
comparable to current statistical uncertainties under the nonparametric
equation of state distributions.
