We study the dissipative thermodynamics of vortex lines in layered sup
erconductors within a simple string model in the dilute limit of negli
gible vortex interactions and compute the specific heat C-upsilon in t
he presence of arbitrary dissipation. The interplay of dissipation, in
ertia, and elasticity is shown to control the qualitative thermodynami
cal behavior and their relative amount determines two very distinct re
gimes for the specific heat. In the dissipation-dominated case we find
a behavior C-upsilon similar to root T for a large interval of temper
ature below T-c.