Suppose I am operating in an environment where Tx and Rx are static and channel does not change between them. In this case if I have channel estimate between Tx and Rx at time T, can I use the same channel estimate at a future time T+tau (assuming tau is very small compared to channel coherence time) for decoding the data. What are the pros and cons ?
If the channel is static, then the coherence time is ideally infinite ( or atleast as long as it stays the same in your case). Channel estimation should be done once for every coherence interval (assuming that this interval is constant-which is not in real channels); else your estimates will be off (this sets the Lower Bound for generating a newer estimates). You can generate a new set of estimates more frequently, but the net information gain would be zero as the channel has stayed same (even if you a newer set of pilots/reference signals) during coherence time and thus it reduces the efficiency of the overall system (vaguely sets Upper Bound).
In short Yes, you may use the same channel estimates.
PS: Remember that this holds true iff everthing else is held constant (Co-channel Interference, Shot/Thermal Noise generated from H/W, and any other H/W Impairments etc).