Runtime Verification (RV) can help find bugs by monitoring program executions against formal properties. Developers should ideally use RV whenever they run tests, to find more bugs earlier. Despite tremendous research progress, RV still incurs high overhead in (1) machine time to monitor properties and (2) developer time to wait for and inspect violations from test executions that do not satisfy the properties. Moreover, all prior RV techniques consider only one program version and wastefully re-monitor unaffected properties and code as software evolves. We present the first evolution-aware RV techniques that reduce RV overhead across multiple program versions. Regression Property Selection (RPS) re-monitors only properties that can be violated in parts of code affected by changes, reducing machine time and developer time. Violation Message Suppression (VMS) simply shows only new violations to reduce developer time; it does not reduce machine time. Regression Property Prioritization (RPP) splits RV in two phases: properties more likely to find bugs are monitored in a critical phase to provide faster feedback to the developers; the rest are monitored in a background phase. We compare our techniques with the evolution-unaware (base) RV when monitoring test executions in 200 versions of 10 open-source projects. RPS and the RPP critical phase reduce the average RV overhead from 9.4x (for base RV) to 1.8x, without missing any new violations. VMS reduces the average number of violations 540x, from 54 violations per version (for base RV) to one violation per 10 versions.