We define the local acceleration and jerk of a relativistic brane in an ambient spacetime, and construct from them a dimensionless parameter lambda that must be small for an interpretation of brane acceleration as local Unruh temperature. As examples, we discuss (i) open rotating branes, for which lambda>1 (ii) closed spherical branes expanding in Minkowski spacetime, for which lambda=0 when the worldvolume is either an Einstein static universe or de Sitter space, in which case the brane temperature equals the Gibbons-Hawking temperature, (iii) closed spherical branes in anti-de Sitter spacetime, for which a maximally symmetric worldvolume is anti-de Sitter, Minkowski or de Sitter according to whether the magnitude of the brane acceleration is less than, equal to or greater than a `critical' value, and (iv) the BTZ black hole, viewed as a membrane.

32 pages, small corrections