For a generalized one-dimensional diffusion, we consider the measure weighted and normalized by a non-negative function of the local time evaluated at a parametrized family of random times, which we will call a clock. The aim is to give a systematic study of the penalization with the clock, i.e., its limit as the clock tends to infinity. We also discuss universal σ-finite measures which govern certain classes of penalizations, thus giving a path interpretation of these penalized processes.