Sorry about that! It's been hard to design an app that is used all over the world. (In fact, we have users regularly viewing the readings from all corners of the globe, including the US Military [i.e., deployed overseas].)
The readings get sent out -- and the RSS feed updated -- at about 06.05 (i.e., in the morning) UTC (or GMT). The readings are sent for the next calendar day; most of our readers asked to get the readings early, so that they could pray and meditate in preparation for the coming day.
At the moment 6 am UTC is 10 pm of the previous day in Anchorage. But in New Zealand, in summer time, it is already 7 pm (of the day after the one in Anchorage!).
The readings will stay in the app's cache for quite a while, so if you look at the readings ahead of time, but want to use them on their correct liturgical day, they will always be available.
I am constrained by the RSS protocol and the application design that uses it at the moment. In the future, I hope to have an app, in which you can choose the date that interests you, and the readings and prayers will be available. However, it will be a bigger app (it must contain all the readings), and I'll have to seek copyright permission because of the quantity of any single translation, so it will be an expensive venture.

At the moment, I'd suggest looking at the readings any time before 10 pm your time, and then you'll have them on the device for the next day. Just scroll down and you will find them.
I am keen for any advice or requests for version 2. So if you have any, please let me know. I'll take the request for a timezone-appropriate version as suggestion No. 1.
