  We investigate some consequences of a specific non-conservation of the
energy-momentum tensor on the physics of certain cosmic string configurations.
This non-conservation is induced by a new gravitational theory recently
introduced as an attempt to incorporate dissipative systems in the gravity
description. This model of gravity is endowed with a variational principle,
where an action-dependence on the geometric sector is assumed. For the
Abelian-Higgs string we derive the dynamical equations and provide the
numerical solutions describing the behavior of the Higgs and gauge fields as
well as the profile of the metric functions for different cases of the
parameter that characterises the model. Additionally, from this numerical
approach we find how the inner structure of the cosmic string is affected by
this underlying gravitational theory, by analyzing the deviations both on the
mass per unit length and the deficit angle of the string. Next, we proceed with
an analytical treatment obtaining the solution close to the string axis and
also the corresponding vacuum solution. Our analytical results are also
complemented with the study of a thick cosmic string model, where the defect is
endowed with finite core. In this case we have obtained the proper interior
solution which joined together with the vacuum metric outside provides an exact
solution for the exterior geometry which generalizes previous results and leads
to a decreasing of the mass per unit length of the cosmic string.
