If someone was trying to gain Gavilar's trust/obedience they wouldn't need to impersonate the Stormfather to do it. I think it's far more likely SF has changed a bit in the last years than this being an impersonation. 

In the prologue for Stormlight Archive book five we see an entity that is referred to as the Stormfather, but which behaves very differently from the Stormfather we have seen in previous books.  This has led people to speculate whether the Stormfather in the prologue is the same spren we've seen in Stormlight 1-4. 

Will resolve to Yes if it turns out that the Stormfather from the prologue is actually Ishar, Odium, Tanavast, one of the unmade or any Spren or person other than character we know as the Stormfather.

Jun 21, 2:20pm: That should read "Will resolve to No if it turns out that...."