sudo -s <<EOF
  a='var'
  echo 'Running serveral commands with sudo'
  mktemp -d
  echo "\\$a"
EOF

• $a needs to be escaped to prevent it to be expanded by the current shell

Or

sudo -s <<'EOF'
  a='var'
  echo 'Running serveral commands with sudo'
  mktemp -d
  echo "$a"
EOF