Hi,
with an Establishment Card you are not allowed to receive unemployment benefits for new graduates (dimittenddagpenge) or any public assistance under the terms of the Active Social Policy Act (lov om aktiv socialpolitik) during your stay in Denmark.
However its allowed to receive “normal” Unemployment benefits (“dagpenge”) as salaried A-kasse member. If you are paid salary as PhD student you can sign up in A-kasse as salaried individual and earn right to dagpenge (see requirements).
BUT as an non EU citizen it only gives meaning to pay for A-kasse if you will have the right to work and stay in Denmark after loosing a job. Your residence Card must allow you to take any job with one day notice. A residence/work permit limited to one job/one employer is not enough. We do not guide about different resident Schemes, i.e. Establishment Card. You musk ask the responsible Authority (SIRI) whether you have right to take any job with one day notice.
In the 6 months job search period you are only allowed to look for and apply for jobs, you are not allowed to work fulltime. So therefore you can not get Unemployment benefits in the 6 months job-search period.
As always you should also contact an A-kasse, and get their opinion about your options.
Let me know if you have further questions.
Best regards,
Anders Weber, A-kasser.dk