Circumplanetary discs can be found around forming giant planets, regardless of whether core accretion or gravitational instability built the planet. We carried out state-of-the-art hydrodynamical simulations of the circumplanetary discs for both formation scenarios, using as similar initial conditions as possible to unveil possible intrinsic differences in the circumplanetary disc mass and temperature between the two formation mechanisms. We found that the circumplanetary discs’ mass linearly scales with the circumstellar disc mass. Therefore, in an equally massive protoplanetary disc, the circumplanetary discs formed in the disc instability model can be only a factor of 8 more massive than their core-accretion counterparts. On the other hand, the bulk circumplanetary disc temperature differs by more than an order of magnitude between the two cases. The subdiscs around planets formed by gravitational instability have a characteristic temperature below 100 K, while the core-accretion circumplanetary discs are hot, with temperatures even greater than 1000 K when embedded in massive, optically thick protoplanetary discs. We explain how this difference can be understood as the natural result of the different formation mechanisms. We argue that the different temperatures should persist up to the point when a full-fledged gas giant forms via disc instability; hence, our result provides a convenient criterion for observations to distinguish between the two main formation scenarios by measuring the bulk temperature in the planet vicinity.