AT and creating a website with distinct sections

Depending on how maintainable you need it to be, you could possibly get what you want by adding a single preprocess HTML hook to theme's THEME_NAME.theme (D8) or template.php (D7) file.

function [THEME_NAME_HERE]_preprocess_html(&$variables) {

  $request_url = $_SERVER['REQUEST_URI'];
  $site_section = 'default';

  if (strpos($request_url, '/about/') !== false) {
    $site_section = 'about';
  }
  elseif (strpos($request_url, '/contact/') !== false) {
    $site_section = 'contact';
  }
  elseif (strpos($request_url, '/store/') !== false) {
    $site_section = 'store';
  }

  $variables['site_section'] = $site_section;

}

Then use that variable in your page/html template to switch out parts of the layout or add it as a CSS class on a top-level element (body?) to recolor any children elements as needed.

A module that lets you manage the sections would be better, but if the sections are set in stone, this might work.

Sorry, I haven't found anything like that in an existing contrib module yet - and nothing directly related to AdaptiveTheme.

Just spitballing ideas: In Drupal 7, it was possible to create an AdaptiveTheme sub-subtheme. AT sub-subthemes only contain the differences with the theme they're based on. If the only problems with ThemeKey are server space and the pain of managing redundant code, sub-subthemes could solve that problem. I haven't had to do that in Drupal 8 yet, though, so I'm just assuming that's still a feature.

Alternatively, the code mentioned above could be turned into a maintainable module using configuration entities, if a custom module is a path you want to explore.

If you're still looking for a contrib solution, you may need to cast a wider net outside the AT issue queue, as that solution wouldn't be limited to AdaptiveTheme. This does feel like a problem that would need to be solved for a lot of sites.

== EDIT ==

It looks like the "Skin base theme" option in the AT Theme Generator does what I was trying to describe with "sub-subthemes"

Skins are sub-sub themes that inherit extension, layout and color settings from their base theme. Select an existing theme as the skins base theme.

Try creating your main theme, create a few skins off of it and see if ThemeKey will use those :)

Depends what you mean by "sections", some details there would help.

Try to give an example, e.g. is this a sections:http://www.aspergillus.org.uk/jobs (plus all the job nodes in that section etc etc?)

ThemeKey is mega powerful on D7, so yes it's an option for sure, if it can provide the conditions that match your site/requirements.

Use light weight skin type themes, really they only need to be a css file (basically) that uses your default theme as a base theme.

This may be a little late to the game, but have you looked at the module Page Theme? I'm using that to apply a different (than the default) AT sub-theme to one content type. I'm happy with it.