  We study the dissipative thermodynamics of vortex lines in layered
superconductors within a simple string model in the dilute limit of negligible
vortex interactions and compute the specific heat $C_v$ in presence of
arbitrary dissipation. The interplay of dissipation, inertia and elasticity is
shown to control the qualitative thermodynamical behavior and their relative
amount determines two very distinct regimes for the specific heat. In the
dissipation dominated case we find a behavior $C_v \sim \sqrt{T}$ for a large
interval of temperature below $T_c$.
