  Poly-Cauchy numbers with level $2$ are defined by inverse sine hyperbolic
functions with the inverse relation from sine hyperbolic functions. In this
paper, we show several convolution identities of poly-Cauchy numbers with level
$2$. In particular, that of three poly-Cauchy numbers with level $2$ can be
expressed as a simple form. In the sequel, we introduce the Stirling numbers of
the first kind with level $2$
