Symptoms
K2 documentation covers installation of K2 for SharePoint for distributed environments but does not detail whether one K2 server with multiple SharePoint 2013 farms scenario implies creation of separate app domains for each SharePoint 2013 farm. You may see errors when installing K2 for SharePoint 4.6.9 on a second farm in case you are trying to install it using the same app domain as you used for the first farm.
Diagnoses
It is necessary to create separate app domain for each SharePoint 2013 farm and it is also necessary to run "K2 for SharePoint Application Deployment.exe" file from the K2 for SharePoint installation folder on SharePoint Central Administration server (in case of a Farm / NLB setup, it should be run on one of the SharePoint servers).
In general for SharePoint-hosted apps you need one domain per farm as otherwise you cannot setup DNS to direct traffic to the respective WFE servers in different farms. You can use any names for domains, e.g. mydomainapps1.com and mydomainapps2.com, they do not need to be descriptive. Refer to Microsoft TechNet documentation for details:
https://technet.microsoft.com/en-us/library/fp161236.aspx
In theory you can use the same app domain for multiple farm with some smart network load balancer (NLB) that can route traffic based on different app prefixes and this can eliminate the need for more than one wildcard SSL certificate.
Provider-Hosted apps do not depend on the app domain. But in case you are adding, for example, a list or page to provider-hosted app manifest later, you will automatically end up with a hybrid between "provider-hosted" and "SharePoint-hosted", and you will in fact have both an "app web" and a "remote web". This is likely to happen to provider hosted app as you add more features to it over time (there are lots of reasons to add a SharePoint hosted list to a Provider hosted app, e.g. to store some state) and this is another factor because of which you should always have a different app domain for every farm.
Resolution
It is necessary to create separate app domain for each SharePoint 2013 farm and it is also necessary to run "K2 for SharePoint Application Deployment.exe" file from the K2 for SharePoint installation folder on SharePoint Central Administration server (in case of a Farm / NLB setup, it should be run on one of the SharePoint servers).
Link to K2 for SharePoint installation documentation for further reference:
http://help.k2.com/onlinehelp/k2forsharepoint/icg/current/default.htm_Install_Topics/OnPrem/On-Premises_Installation.htm?TocPath=_____4
