  We consider the damped wave equation on a compact manifold. We propose
different ways of measuring decay of the energy (time averages of lower energy
levels, decay for frequency localized data...) and exhibit links with resolvent
estimates on the imaginary axis. As an application we prove a universal
logarithmic lower resolvent bound on the imaginary axis for the damped wave
operator when the Geometric Control Condition (GCC) is not satisfied. This is
to be compared to the uniform boundedness of the resolvent on that set when GCC
holds. The proofs rely on (i) various (re-)formulations of the damped wave
equation as a conservative hyperbolic part perturbed by a lower order damping
term;(ii) a "Plancherel-in-time" argument as in classical proofs of the
Gearhart-Huang-Pr{\"u}ss theorem; and (iii) an idea of Bony-Burq-Ramond of
propagating a coherent state along an undamped trajectory up to Ehrenfest time.
