  Cooling of particles to mK-temperatures is essential for a variety of
experiments with trapped charged particles. However, many species of interest
lack suitable electronic transitions for direct laser cooling. We study
theoretically the remote sympathetic cooling of a single proton with
laser-cooled $^9$Be$^+$ in a double-Penning-trap system. We investigate three
different cooling schemes and find, based on analytical calculations and
numerical simulations, that two of them are capable of achieving proton
temperatures of about 10 mK with cooling times on the order of 10 s. In
contrast, established methods such as feedback-enhanced resistive cooling with
image-current detectors are limited to about 1 K in 100 s. Since the studied
techniques are applicable to any trapped charged particle and allow spatial
separation between the target ion and the cooling species, they enable a
variety of precision measurements based on trapped charged particles to be
performed at improved sampling rates and with reduced systematic uncertainties.
