The base of the FreeAgent drive contains the bridge electronics which handle communication with the computer. It is normal for this area to be warm. The base of the drive does not contain the hard drive, so a warm base is not likely to cause a drive failure.
If the base is merely warm, that is an indication of normal operation.
Be sure that nothing is blocking the vents on the bottom of the drive, as that is how the electronics in the base cool themselves. The drive is designed to sit on a flat surface like a desk, so if it has been placed on carpet or some other surface that may be blocking the vents, the drive should be moved.
If the unit is properly ventilated and the base is extremely hot, please create a warranty replacement order at our Warranty Services Page as the unit may be malfunctioning.
As for the electronics being warm when the unit is powered off, this is also normal. This drive has advanced power management features whereby the device monitors the bus (ie, the cabling - USB/1394/eSATA) for activity. During periods of inactivity, the drive is spun down, but the electronics in the base stay active to monitor for activity so the drive will respond quickly. It also monitors for signal.
If the computer is turned off (and the signal is terminated), then the drive spins down, but the electronics in the base continue to monitor for a new signal, so that when the computers turns on, the drive will spin up again.