One of the most important questions, especially in applications, is how to choose a decay function in the study of stability for a concrete equation. Motivated by the fact that the coefficients of the considered equation mainly suggest the choice of the decay function, the point of analysis in the paper is to carry out the Lyapunov function approach and to state coercivity conditions dependent on decay in the study of the $p$th moment stability with a general decay rate for a certain stochastic differential equation with variable time delay. Some criteria including the usual exponential decay are discussed, particularly the ones with a concave decay, special cases of which are polynomial and logarithmic decays. Some consequences and examples are given to illustrate the theory.