I used Chrome Web Inspector to look at how cloud.digitalocean.com is using cookies to keep you logged in.
You can see that DigitalOcean uses a SESSION cookie to maintain your logged-in state. If you read about session cookies, you will find that different browsers, OSes, and user-behaviors result in very different session cookie lifetimes.
For example, on CHROME, as long as you never completely shut down the Chrome browser process, your session cookie will be maintained. If you are running Chrome in OSX, closing all the browser windows won’t necessarily close Chrome completely, meaning your Session cookies will be saved. This is the intended behavior
On Firefox: The “Restore previous session” feature will also restore SESSION cookies that were set in your last browsing session, you can change the settings so that this happens by default every time you start up Firefox too. Just keep in mind that may create other privacy issues.
TL;DR: DigitalOcean uses session cookies to maintain logged-in state, there are browser settings you can change to maintain session cookies, but this may create other privacy issues.