It depends what is the contract on $this->currentDatabase. Can it change at any time, after any method call? If it changes, are you supposed to keep on using the object you did when you made your first db call, or are you supposed to always us the current value? This dictates if you must always use $this->currentDatabase, or if you must always store it in a variable before using.
So, strictly speaking, this is not a style question at all.
But, assuming the member is never changed during function calls such as this, it makes no difference. I'd say storing it in a variable is slightly better, as it is easier to read and avoids a member access on an object at every operation. The compiler may optimize it away if it's good, but in many languages such optimizations are very difficult - and accessing a local variable is almost invariably faster than accessing a member of an object.