Speed is the length of travel per unit of duration (or time interval). Spatial rest is a speed of zero. That is, there is no change in location per unit of time. A body does not change location (relative to an inertial observer) while time continues.
But temporal rest seems different. It cannot be zero pace because that would mean it takes no time to go a positive distance, right? No, that is not what zero pace means.
Pace is the travel time per unit of length (or stance interval). Time is the dependent variable and length is the independent variable.
Consider a race that is about to begin. The runners are in place waiting for the signal to start. The official timer is set to begin. In terms of motion, the runners are at rest with speed of zero. They are not making any distance, but time continues as usual.
What is the pace of the runners in that case? There is no change on the official timer. But the stance continues as usual. For example, if stance is related to the distance from the Sun of a Voyager spacecraft (see here), it continues to increase as usual.
A map with a time scale shows a point for a pace of zero. Despite the distance made by an odologe, a body with a pace of zero remains in the same place in time. It is at rest in time.
What about an infinite value for pace in time? The Galilean transformation implicitly has an infinite speed of information for space, which makes information spatially ubiquitous since it travels an infinite distance for a finite time. The symmetric Galilean transformation implicitly has an infinite pace of information in time, which makes information temporally ubiquitous since it takes an infinite time to travel a finite distance.