Acrobat is prompting you to save the PDF to temp folder, as it downloaded to this location whenever you click the link in IE.
To help you fix the issue, I have some questions:
1. What browser are you using and its version.
2. Please check whether "Adobe Acrobat Sharepoint OpenDocuments Component" and "Adobe PDF Reader" add-ins are enabled in IE or not. If not, please enable them and then try to open the file.
3. If it still doesn't work for you, can you please open DocIcon.xml from %ProgramFiles%\CommonFiles\MicrosoftShared\webserverextensions\15\TEMPLATE\XML and check if there is entry for "pdf"(Mapping Key="pdf"). If yes, can you please share the whole XML entry here.