Creating custom abstract environment

I don't have the kaobook class but below I used scrbook which you said was the basis for kaobook. The following revises your secabstract environment to set text in italics.

% abstractprob.tex  SE 571787
documentclass{scrbook}
usepackage{lipsum}

providecommand{abstractname}{Abstract} % not in scrbook class
newenvironment{secabstract}[1]{%
hrule
smalltextbf{abstractname: } 
%smallemph #1     %  emph takes an argument
smallemph{#1} % or smalltextit{#1} 
itshape % use this if you want the text to be in italics
}{%
newlinehrule
vspace{0.6cm}
}

begin{document}
begin{secabstract}{An abstract}
lipsum[1]
end{secabstract}
end{document}