Continuity in the time parameter is considered for a natural version of the local time for stationary Gaussian processes. Bounds are given for the local and uniform modulus of continuity which are applicable in cases not covered by Kono, notably when the incremental variance is a regularly varying function of index two.
"Continuity of Gaussian Local Times." Ann. Probab. 10 (3) 818 - 823, August, 1982. https://doi.org/10.1214/aop/1176993790