I do not understand well what you mean by Killing horizon here. A Killing horizon in a 4-dimensional spacetime is, by definition, a null 3-surface made of integral lines of a Killing field defined in a neighbourhood of that surface, which becomes null exactly on that surface. As is known, the surface gravity turns out to be always constant along each integral line (which can be proved to be geodesic).
I guess that actually you are wondering if the surface gravity must be constant also changing the integral curve of the Killing field while remaining on the horizon.
It is possible to prove that, whenever a spacetime admitting a Killing horizon satisfies Einstein equations and the dominant energy condition is verified, the surface gravity must be constant on
the whole horizon.
However a result, originally obtained by Carter, states that the surface gravity of a Killing horizon must be constant if (i) the Killing field is static or (ii) there is an additional Killing
field and the two Killing fields are 2-surface orthogonal.
All that can be found in Wald's textbook on general relativity. Now I cannot check the precise page, but I am sure on that.