Reduce BT power consumption by dis- and reconnecting
I do not believe there is a simple answer because it depends on the hardware.

Let's assume that you have an IC that removes the complexity of the communications, such as a BroadComm BCM43012.  These ICs already have software internally that turn off/on the radio and this abstracts the concerns of quiescent current draw of the analog from the interface.  When you "connect", you actually have a bit of delay as the VCOs come up with the rest of the support circuits.

The assumption is that the IC can handle the power management better than the programmer, and this is generally true when combined with the OS level.  Often, the OS is handling the power management of the wireless ICs if the IC itself is not.