In general I think Harrison's paper falls into the trap of treating the FLRW geometry as a sort of background on which galaxies move as test particles, when in reality it's the gravitational field of the galaxies. That's a critical error when talking about conservation of energy, because treating the FLRW geometry as a background means treating the matter in the universe as exerting a gravitational influence on your test particles but not being gravitationally influenced by them, i.e., ignoring Newton's third law. Energy seems to mysteriously appear and disappear because he isn't modeling the dynamics of the full system.
Gott and Rees say that the density of cosmic strings decreases as $a^{-2}$ rather than $a^{-3}$. I think this is also an error with a somewhat similar origin.
The argument for $a^{-2}$ is that if you have a volume of space crisscrossed by cosmic strings, and the scale factor doubles, the total length of the strings doubles; since their linear energy density is constant, their total energy doubles; and the volume of space increases by a factor of $8$, so the density gets an overall factor of $1/4$.
That's correct if the strings are made of noninteracting dust that moves with the Hubble flow, but in reality they're dynamical objects that behave like rubber bands under tension (except that the tension is independent of the total length, like QCD flux tubes). They would likely pick up an initial outward motion from whatever process produced the initial conditions of FLRW cosmology, and so they would expand at first, but only until their internal tension reversed the expansion and they collapsed again (this is independent of the overall expansion/collapse of the universe). The kinetic energy of their initial expansion contributes to their mass, so when they expand, the extra energy implied by the extra length doesn't appear out of nowhere; it comes from the kinetic energy. You can't neglect these effects globally. If you neglect them locally, you're treating the region of space under consideration as anomalous, and you get a funny scaling law because the apparent local scale factor doesn't match the true global scale factor.
