Note that calling $apply will trigger $digest on the $rootScope, which means $digesting all child scopes.
The $apply function itself is relatively light (you can inspect it in the angular source). It is the process of evaluating watchers and comparing values (dirty-checking) during a $digest that can get expensive. Because of this, performance testing to date has been focused on measuring the $digests. Some examples:
How does data binding work in AngularJS? (see Misko's answer)
How Do I Measure the Performance of my AngularJS app's digest Cycle?
http://blog.scalyr.com/2013/10/angularjs-1200ms-to-35ms/
This is a great explanation of the difference between an $apply and a $digest: http://www.sitepoint.com/understanding-angulars-apply-digest/. Relevant excerpts:
$digest:
The $digest cycle starts as a result of a call to $scope.$digest(). Assume that you change a scope model in a handler function through the ng-click directive. In that case AngularJS automatically triggers a $digest cycle by calling $digest(). When the $digest cycle starts, it fires each of the watchers. These watchers check if the current value of the scope model is different from last calculated value. If yes, then the corresponding listener function executes. As a result if you have any expressions in the view they will be updated. In addition to ng-click, there are several other built-in directives/services that let you change models (e.g. ng-model, $timeout, etc) and automatically trigger a $digest cycle.
$apply:
AngularJS ... will account for only those model changes which are done inside AngularJS’ context (i.e. the code that changes models is wrapped inside $apply()). Angular’s built-in directives already do this so that any model changes you make are reflected in the view. However, if you change any model outside of the Angular context, then you need to inform Angular of the changes by calling $apply() manually. It’s like telling Angular that you are changing some models and it should fire the watchers so that your changes propagate properly.