Do lithium-burning brown dwarfs (65+ Jupiter masses) have starspots too or do they occur only on hydrogen-burning stars?
Brown dwarfs occupy the gap between the highest mass planets and the lowest mass stars. They do not attain internal temperatures high enough to ignite the nuclear burning of hydrogen, but they are typically thought to be able to burn lithium.
Starspots are regions on the surface of a star that are visibly darker than the surrounded areas because it is cooler. The current paradigm for understanding the emergence of starspots is magnetic reconnection:
IT is possible that in the future, infrared observations could be used to measure the polarimetry of a brown dwarf's surface, as it is done for the Sun's surface, to try to look for sunspots, since seeing them visually will be very difficult since brown dwarfs radiate in infrared, predominantly. (And since the starspot is darker region of the surface, it'll be even harder to see visually).
Solar flares from brown dwarfs have been observed for a couple decades, which helped to reinvigorate studies into the internal dynamics of brown dwarfs, along with other discoveries. This field is very active today, for example nearly a hundred brown dwarfs have been discovered near our Sun (I accidentally said galactic center here earlier, my bad!), and with the prospect for things like the LISA mission to observe them near the galactic center makes the future very bright for this field!
Anyway, since solar flares are thought to be a result of starspot activity, observations of brown dwarf solar flares indicate that there might be starspots on the surfaces of brown dwarfs. This would not be totally surprising since brown dwarfs are thought to typically harbor strong magnetic fields: "Because it has no strong central nuclear energy source, the interior of a brown dwarf is in a rapid boiling, or convective motion. When combined with the rapid rotation that most brown dwarfs exhibit, convection sets up conditions for the development of a strong, tangled magnetic field near the surface." This site also has some great content about observed flares from brown dwarfs.
There are high mass brown dwarfs that are observed to have very strong magnetic fields, which, in the magnetic reconnection paradigm, is required for starspots to emerge. This implies that, together with obseravtions of solar flares, it would not be totally surprising to find starspots on brown dwarfs.
Theoretically, simulations of the emergence, evolution, and stability of magnetic fields in brown dwarfs are very uncertain currently - hence the need for more observations! - but these simulations suggest that the strength of the dwarf's magnetic field depends strongly on deuterium burning.
For example, in this study they assume the dwarfs are rotating fast, and find that "massive brown dwarfs of about 70 Jupiter masses can have fields of a few kilo-Gauss during the first few hundred Million years." The fields weaken over time after deuterium burning ceases, but such field strengths are likely sufficient for producing starspots, since the Sun's starspots are on the order of kilo-Gauss as well.
If we observe these features on brown dwarf, then they might need a more general name than "starspot," perhaps "darkspot" or something would be better? Idk ;)