Room temperature is fine.  It'll keep for years.

There is an anecdotal story about a partially filled bottle of Rodinal found
in the ruins of a German city in the aftermath of WWII.  The developer was
tested and found to be perfectly good after those many years of exposure to
the elements.

If the plastic containers now used give you pause, store them in an airtight
container - decanting into the proverbial brown glass bottles can work, as
can placing the developer as is in those airtight used ammunition metal
boxes.

As with all chemicals, I imagine that a cool, temperate, dark storage area
would work best.  A fridge is not needed - perhaps a wine storage area or
even a dark closet with little temperature fluctuations, under some
insulation (sleeping bags, old blankets, etc.).

A general rule of thumb is that chemical reactions double
in speed every 10 C.

By this criteria if you chill the Rodinal by 20 C/36 F it
should last 4 times as long.  Rodinal is known to be still
good after 20 years storage at room temperature -> it should
last 80 years in the 'fridge.  [-> being the Murphean symbol
for "very unlikely"].

OTOH, some of the ingredients may crystallize out when it is
chilled.  OYAH, if this happens it may be that a little bit of
warming is all that's needed to restore it.

Take a small vial of the stuff, chill it and see what happens.

add marbles to the bottle as you use the developer, keeping the bottle
full for a much longer and eventually transfering the remaining
developer to a smaller bottle.

Bob McCarthy