Slupring the whole file into memory at once tends to often suck. And avoiding that also means you can avoid complicated regexes (using a very SMoP instead).

my $message= '';
while( <> ) {
    if( ! /^\d{4}-\d\d-\d\d / ) {
        $message .= $_;
    } else {
        print $message
            if(  $message  =~ /keyword/  );
        $message= $_;
    }
}
print $message
    if(  $message  =~ /keyword/  );
[download]

Or, if you prefer:

my $line= <>;
while( defined $line ) {
    my $message= '';
    do {
        $message .= $line;
    } while(    defined( $line= <> )
            &&  $line !~ /^\d{4}-\d\d-\d\d /
    );
    print $message
        if(  $message  =~ /keyword/  );
}
[download]