  Appearance of cosmic strings in the early Universe is a common manifestation
of new physics typically linked to some high energy scale. In this paper, we
discuss a different situation, where a model underlying cosmic string formation
is approximately scale free. String tension is naturally related to the square
of the temperature of the hot primordial plasma in such a setting, and hence
decreases with (cosmic) time. With gravitational backreaction neglected, the
dynamics of these melting strings in an expanding Universe is equivalent to the
dynamics of constant tension strings in a Minkowski spacetime. We provide an
estimate for the emission of gravitational waves from string loops. Contrary to
the standard case, the resulting spectrum is markedly non-flat and has a
characteristic falloff at frequencies below the peak one. The peak frequency is
defined by the underlying model and lies in the range accessible by the future
detectors for very weak couplings involved.
