This is a good question.
At first glance, it does not appear to be necessary to relate every variable with time. It helps to distinguish between variables in modelling a physical system (call them physical variables) and abstract variables in non-physical systems.
Physical variable. A variable is a symbol on whose value a physical system function depends. For example, the entropy H({temp(t)}) of a collection of randomly changing readings temp(t) of temperature probes in a large reservoir that varies over time t.
Abstract variable. A variable is a symbol on whose value a function, polynomial, etc., depends. Stover, Christopher and Weisstein, Eric W. "Variable." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Variable.html
For example, let p(x) = the probability of the occurrence of x and h(x) is the information content of x, which is defined to be log_2 1/p(x). In this case, x is the value of a random variable and h(x) has been defined without reference to time.
That is, a random variable as a real function whose domain is the probability space and such that:
1. The set {X
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