  We construct models of slowly rotating, perfect-fluid neutron stars by
extending the classical Hartle-Thorne formalism to generic scalar-tensor
theories of gravity. Working at second order in the dimensionless angular
momentum, we compute the mass M, radius R, scalar charge q, moment of inertia I
and spin-induced quadrupole moment Q, as well as the tidal and rotational Love
numbers. Our formalism applies to generic scalar-tensor theories, but we focus
in particular on theories that allow for spontaneous scalarization. It was
recently discovered that the moment of inertia, quadrupole moment and Love
numbers are connected by approximately universal (i.e., equation-of-state
independent) "I-Love-Q" relations. We find that similar relations hold also for
spontaneously scalarized stars. More interestingly, the I-Love-Q relations in
scalar-tensor theories coincide with the general relativistic ones within less
than a few percent, even for spontaneously scalarized stars with the largest
couplings allowed by current binary-pulsar constraints. This implies that
astrophysical measurements of these parameters cannot be used to discriminate
between general relativity and scalar-tensor theories, even if spontaneous
scalarization occurs in nature. Because of the well known equivalence between
f(R) theories and scalar-tensor theories, the theoretical framework developed
in this paper can be used to construct rotating compact stellar models in f(R)
gravity. Our slow-rotation expansion can also be used as a benchmark for
numerical calculations of rapidly spinning neutron stars in generic
scalar-tensor theories.
