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There is an open standard - it's called Web Push (https://www.w3.org/TR/push-api/). It's already used by Firefox and Chrome to implement push notifications for web pages (via service workers). The notifications are proxied via Mozilla/Google server respectively, but there is no reason you could not run your own server.

It's not an open standard, but this was the one of the interesting pieces of RIM's software stack. Their push system worked reasonably well, used little power on the handset, and organizations were able to run their own on-site push proxy supporting their own in-house applications.

BlackBerry cracked push like this years ago, though it wasn't open. The phones used to connect over the carrier's network to a private APN (over VPN or leased line with to the carrier), so they always knew your BlackBerry's IP address. If you were out of range they'd do the whole storing it all up and when your device came back into coverage they'd send it all down. No hanging connections, the device was directly exposed into the BlackBerry network.