How to detect if the current page is the homepage with CakePhp?

Simply you can try this:

if ($this->request->here == '/') {
       // some code
}

Also it is good to read this part of documentation:

You can use CakeRequest to introspect a variety of things about the request. Beyond the detectors, you can also find out other information from various properties and methods.

$this->request->webroot contains the webroot directory.
$this->request->base contains the base path.
$this->request->here contains the full address to the current request
$this->request->query contains the query string parameters.

You can find it by comparing the present page with webroot or base

if ($this->here == $this->webroot){ // this is home page }

OR

if ($this->here == $this->base.'/'){ // this is home page }