Can you send me the link to those rules? Thanks.
AND RULES OF THE HOUSE OF REPRESENTATIVES
https://www.govinfo.gov/content/pkg/HMAN-116/pdf/HMAN-116.pdf
Page 372
Rule 1 Clause 8
(As far as limited powers for the Speaker Pro Tem there are limitations but only when he is serving due to illness of the speaker)
(b)(1) In the case of illness, the Speaker may appoint a Member to perform the duties of the Chair for a period not exceeding 10 days, subject to the approval of the House.
(3)(A) In the case of a vacancy in the Office of Speaker, the next Member on the list described in subdivision (B) shall act as Speaker pro tempore until the election of a Speaker or a Speaker pro tempore. Pending such election the Member acting as Speaker pro tempore may exercise such authorities of the Office of Speaker as may be necessary and appropriate to that end.
(Legal scholars are going nuts right now interpreting what that means but on face value, it looks to me like the Pro Tem has the authority of the speaker. However, I did not stay at a Holiday Inn Express last night sooo....)