Allow authenticated users to vote multiple times with Rate module

With Rate module you can use this solution (this code is part of my custom module - in my case it's name is regions_vote - i hope you get idea):

define('REGIONS_VOTE_LIMIT', 60); // 60 seconds for demonstration

/**
 * Implements hook_rate_vote_alter()
 *
 * @param type $votes
 * @param type $context
 */
function regions_vote_rate_vote_alter(&$votes, &$context) {
  // We want save all votes
  $context['criteria'] = FALSE;

  // User vote allowed only if current time is higher than user
  // latest vote time + limit vote time
  $latest_vote = regions_vote_check_vote_allowed($votes);
  if ($latest_vote['status'] === FALSE) {
    $context['save'] = FALSE;
    $_SESSION['regions_vote'] = $latest_vote;
  }
}

/**
 * Implements hook_rate_widget()
 *
 * @param type $op
 * @param type $widget
 * @param type $values
 */
function regions_vote_rate_widget($op, &$widget, $values = NULL) {
  if ($op == 'view') {
    $_SESSION['regions_vote'] = regions_vote_check_vote_allowed(array(
      'content_type' => $values['content_type'],
      'content_id' => $values['content_id'],
      'tag' => REGIONS_VOTE_RATE_TAG,
    ));
  }
}

/**
 * Preprocess function for the regions_vote template.
 */
function regions_vote_preprocess_regions_vote_template_regions_vote(&$variables) {
  extract($variables);

  $variables['button'] = theme('rate_button', $links[0]['text'], $links[0]['href'], 'rate-regions-vote');

  $info = array();
  if ($mode == RATE_CLOSED) {
    $info[] = 'Closed.';
  }
  if ($mode != RATE_COMPACT && $mode != RATE_COMPACT_DISABLED) {
    if (isset($results['user_vote']) && $mode != RATE_CLOSED) {
      if ($just_voted) {
        $info[] = 'Thanks for your vote. You can add new vote after '. $_SESSION['regions_vote']['new_vote_allowed_rendered'];
      }
      else {
        if ($_SESSION['regions_vote']['status'] === FALSE) {
          $info[] = 'Thanks for your vote. You can add new vote after '. $_SESSION['regions_vote']['new_vote_allowed_rendered'];
        }
        else {
          $info[] = 'Thanks for your vote. You could have added new vote for '. $_SESSION['regions_vote']['new_vote_allowed_rendered'];
        }
      }
    }
    else {
      $info[] = $results['count'] .' votes.';
    }
  }
  $variables['info'] = implode(' ', $info);

  if ($just_voted === FALSE) {
    unset($_SESSION['regions_vote']);
  }
}

/**
 * Check if user can vote now.
 *
 * @global type $user
 * @param type $votes
 * @return type
 */
function regions_vote_check_vote_allowed($votes) {
  global $user;

  // Current time.
  $current_time = time();
  // Only one vote per limit per user.
  $limit_vote_time = REGIONS_VOTE_LIMIT;

  // Get latest vote time for this content for current logged user.
  $latest_vote_time = db_result(db_query(
    "SELECT
       timestamp
     FROM {votingapi_vote}
     WHERE
       content_type = '%s' AND
       content_id = %d AND
       tag = '%s' AND
       uid = %d
     ORDER BY timestamp DESC
     LIMIT 1",
    $votes['content_type'],
    $votes['content_id'],
    $votes['tag'],
    $user->uid
  ));

  // User vote allowed only if current time is higher than user
  // latest vote time + limit vote time
  $new_vote_allowed = $latest_vote_time + $limit_vote_time;
  $new_vote_allowed_in_seconds = $new_vote_allowed - $current_time;

  // User can not vote new_vote_allowed_in_seconds seconds...
  if ($current_time < $new_vote_allowed) {
    return array(
      'status' => FALSE,
      'current_time' => $current_time,
      'new_vote_allowed' => $new_vote_allowed,
      'new_vote_allowed_in_seconds' => $new_vote_allowed_in_seconds,
      'new_vote_allowed_rendered' => format_interval($new_vote_allowed_in_seconds),
    );
  }

  // User can vote now...
  return array(
    'status' => TRUE,
    'current_time' => $current_time,
    'new_vote_allowed_in_seconds' => abs($new_vote_allowed_in_seconds),
    'new_vote_allowed_rendered' => format_interval(abs($new_vote_allowed_in_seconds)),
  );
}