  We have systematically measured the transport properties in the layered
rhodium oxide K$_{x}$RhO$_{2}$ single crystals ($0.5\lesssim x \lesssim 0.67$),
which is isostructural to the thermoelectric oxide Na$_{x}$CoO$_{2}$. We find
that below $x = 0.64$ the Seebeck coefficient is anomalously enhanced at low
temperatures with increasing $x$, while it is proportional to the temperature
like a conventional metal above $x=0.65$, suggesting an existence of a critical
content $x^{*} \simeq 0.65$. For the origin of this anomalous behavior, we
discuss a filling-induced Lifshitz transition, which is characterized by a
sudden topological change in the cylindrical hole Fermi surfaces at the
critical content $x^*$.
