  One-dimensional topological gravity is defined as a Gaussian integral as its
partition function. The Gaussian integral supplies a toy model as a simpler
version of one-matrix model that is well known to provide a description of
two-dimensional topological gravity. The one-dimensional topological gravity
inherits an integrable hierarchy structure as with two-dimensional topological
gravity, yet it is the Burgers hierarchy rather than the Korteweg--de Vries
hierarchy. Making use of this fact, an extension of the one-dimensional
topological gravity to an analogue of two-matrix model is investigated and the
associated partition function is shown to consist of a pair of partition
functions of one-dimensional topological gravity intertwined via the
Moyal--Weyl product, which enables to provide an explicit formula for its free
energy. The extended system shows a hierarchy structure interpreted as a
noncommutative extension of the Burgers hierarchy. The relation to
noncommutative U(1) gauge theory is suggested.
