Solar system objects with perihelia beyond the orbit of Jupiter ($q >$ 5 AU) are too cold for water ice to generate an appreciable coma via sublimation. Despite this, numerous high perihelion objects (HPOs) including many comets and recently escaped Kuiper belt objects (``Centaurs'') are observed to be active out at least to the orbit of Saturn ($q \sim$ 10 AU)... Peak equilibrium temperatures at 10 AU ($\sim$125 K), while far too low to sublimate water ice, are sufficient to sublimate super-volatiles such as CO and CO$_2$ ice. Temperatures at 10 AU are also high enough to trigger the rapid crystallization of exposed amorphous ice, thus constituting another possible driver of distant activity. While supervolatile ices can sublimate strongly (as $r_H^{-2}$) to at least Kuiper belt (30 AU) distances, crystallization is an exponential function of temperature that cannot be sustained much beyond $\sim$10 AU. The heliocentric dependence of the activity thus suggests an observational test. If activity in high perihelion objects is triggered by crystallization, then no examples of activity should be found with perihelia $q >>$ 10 AU. If, on the other hand, activity is due to free sublimation of exposed supervolatile ices, or another cause, then distant activity might be detected. We obtained sensitive, high resolution Hubble Space Telescope observations of HPOs to search for activity beyond the crystallization zone. No examples of activity were detected in 53 objects with $q >$ 15 AU, consistent with the crystallization trigger hypothesis. However, sensitivity limits are such that we cannot reject the alternative hypothesis that mass loss is driven by the sublimation of supervolatile ices. We also searched for binary companions in our sample, finding none and setting an empirical 3$\sigma$ limit to the binary fraction of $<8$\%. (read more)